G'Kar, in "A Late Delivery From Avalon", jumps in to help a human who stepped in to defend a helpless woman, evening the odds in the fight significantly. G'kar observes, in a post-fight drink:
"...and they made a very satisfying thump when they hit the ground!"
In the season 1 episode Signs and Portents, the station is seemingly overwhelmed by Raider ships, then suddenly Sinclair springs his trap when the supposed Starfury squadron the Raiders had lured away returned.
In season 2 All alone in the night, a fleet led by the EAS Agammemnon and flanked by Starfuries and Ambassador Delenn's Flyer, arrive in the nick of time to stop a Streib ship from making off with Captain Sheridan.
The classic Babylon 5Big Damn Heroes moment is when the Minbari arrive in Severed Dreams:
Delenn: "Only one human captain has ever survived battle with a Minbari fleet. He is behind me. You are in front of me. If you value your lives, be somewhere else."
Later in season 3's Walkabout, just as the Shadow ships are closing in on the Whitestar, the Army of Light arrives led by the G'tok.
Also in season 3's Interludes and Examinations, the Brakiri were having their asses kicked by the rampaging Shadow fleet when suddenly, a Vorlon armada jumps in and proceeds to annihilate the enemy fleet and at the same time rally the League of Non-Aligned worlds into an alliance.
In season 4's episode "Endgame", Clark had given a last "scorched earth" order upon his imminent defeat, setting one of Earth's defense platforms to fire on the planet itself. With many ships damaged and no other way to destroy the platform in time, Sheridan on the Agamemnon steers the ship to ram the platform. Just before impact would have happened, General Lefcourt, who had commanded the Earthforce defense against Sheridan's advance on Mars and Earth, becomes a Big Damned Hero for Sheridan when his ship Apollo, which just recovered from extensive disablement in the battle of Mars, jumps in and fires on the platform, destroying it.
Lefcourt: We have monitored the situation. Hold onto your hats!
In season 5's A View from the Gallery, just as the station is about to be overrun by the unnamed invaders, the Whitestar fleet jumps in to save the day.
Deconstructed in Season 5's Learning Curve, when a young Minbari Ranger steps in to protect a human being assaulted in DownBelow. He gets his ass kicked and nearly dies because he's not yet experienced enough to handle himself in such a situation. The rest of the episode is about him regaining his confidence and exploring his own motives in joining the Rangers.
Band of Brothers: In episode 3, the Germans are assaulting positions held by the 101st Airborne outside of the French town of Carentan. While Easy Company is holding their own, several other companies have been forced back by the German attack. Cue the arrival of 2nd Armored. The Germans, caught off guard, are forced into a panicked retreat.
Battlestar Galactica (Reimagined): In a two-part episode of the 2004 reimagining of the show, called "Exodus", the titular ship is attacked by three Cylon Baseships while attempting to help colonists on the planet New Caprica escape. The ship is severely damaged, and somber music plays while the camera zooms out as the Baseships pummel Galactica to death in the cold darkness of space...until a missile, then more missiles, fire off the edge of the screen, and the BattlestarPegasus hurtles to the rescue, just in the nick of time. This is just the high point to heroic events which kicked off with a deadpan"Prepare for turbulence."
In season 4, episode 13, Lee is captured by members of Zarek's rebellion against the human-Cylon alliance. It looks as though he's about to be shot, and there's a gunshot...and you see Lee's face, spattered with blood, and Starbuck standing there, pistol in one hand. Another of the rebels moves, and she draws another pistol, shooting him. "I can do this all day." It escalates the situation later on, but it's awesome at the time.
The classic Original Series had moment after moment of this, with every major and minor character getting a chance to pull one off in one episode after another.
Bones: Averted this in a recent episode, with the FBI showing up to save the day only to find that the "bad guys" had already departed, and the day was no longer in need of saving
Xander: I saved the world with talking from my mouth. My mouth saved the world.
Angel's entrance in Season 7 also qualifies.
"You look timely. And also good."
Averted in Angel Season 5 when Illyria becomes concerned about Wesley and arrives just after he's been fatally wounded. She did take care of his opponent though.
The end of the episode "Double Or Nothing" when the team comes to save Gunn Just in Time.
Castle, of course, has its share. One of the better ones being "The Wild Rover", when Ryan has just revealed his true identity to the boss of a gang he was infiltrating, he actually calls out his heroes, starting with his partner Esposito. Though it takes a second (pause for dramatic effect?) enter Espo...then enter Beckett...then enter Castle (though he's comedically not packing like Espo and Beckett are).
Chuck: In the Season 2 finale, the day is saved in the form of John Casey and his unit parachuting into the Church, to the tune of Mr. Roboto.
Community: In the episode "Epidemiology" when Greendale is overrun by zombies, at the last moment Troy saves the day by cooling everyone down until the military arrived.
Criminal Minds: Does this almost every episode, with the team racing in just before the killer/rapist/terrorist/etc kills his next victim.The team figures out the next target, rushes to the Monster's hideout and saves the victim from certain death. However, they do occasionally subvert this trope (most notably in "Zugzwang") and it is implied that in cases that aren't shown, it happens more frequently.
Doctor Who: Mickey and Jackie showing up to save Sarah Jane in "Journey's End".
Not to mention the Doctor smashing through a mirror astride a white horse to save Madame De Pompadour from the creepy clockwork men in "The Girl in the Fireplace".
Cleverly framed toward the end of "The Doctor Dances". Jack Harkness is sitting in his cockpit, sipping a martini, resigned to the imminent destruction of his ship. The camera pulls away from Jack slowly, showing him in the center as the camera passes through the interior of his ship...and into the TARDIS. Cut to the Doctor and Rose dancing, and Rose shouts to Jack, "Well, c'mon!"
The seventh Doctor and the army rush into a school just in time to save Ace from the Daleks.
Subverted in "The Waters of Mars". The Doctor tries to alter history to save Adelaide Brooks and her remaining crew, but nearly broke time itself instead; this was averted only through Adelaide's self-sacrifice.
Played awesomely straight in 'The Runaway Bride'. Donna is kidnapped by a Robot Santa Taxi Driver and is being driven down a motorway. She looks out of the back window, screaming for help, just in time to see the TARDIS swoop down from the sky and underneath a bridge. It then approaches the Taxi at speed, spinning wildly, after taking a sizable chuck out of the tarmac and nearly flattening some of the traffic. Donna's reaction sums it up to the whole audience:
Donna: You are kidding me.
Rory appearing to save Amy, Amelia and the Doctor from the stone Dalek in "The Big Bang".
Amy returns the favor in "The Wedding of River Song" by gunning down the three Silence about to kill Rory.
Having earlier rescued the Doctor and Holy Roman Emperor Winston Churchill from dozens of Silence hanging from the roof above their heads, this time with the help of both Rory and a platoon of soldiers.
The TARDIS is basically an enforcer of this trope, purposely dropping the Doctor off wherever she knows he's needed (existing across all of time and space comes in handy for figuring this out) so that he'll just "happen" to be in the area whenever things go pear-shaped somewhere in the universe, and will swoop in and save the day.
The Doctor does this in The Snowmen with a Mr. Punch puppet of all things!
Taken to epic proportions in The Day of the Doctor: The current Doctor, all 11 of his past incarnations, as well as his next incarnation play this role for the people of Gallifrey during the climax.
Earlier than that, Three incarnations of The Doctor made a Big Damn Heroes entrance into a room that was completely TARDIS proof by freezing themselves in a piece of Gallifreyan art, arranging for that art to be moved to the room in question, and then sonicing an active Dalek through the picture frame to step in and avert a nuclear holocaust.
In The Day of The Moon, the unmistakable tones of the TARDIS materializing announce the arrival of The Doctor, Rory, and River Song, just in time to save Amy from The Silence.
One particularly fantastic human-led Big Damn Heroes moment occurs in Victory of the Daleks, when right as the Doctor is about to be killed because his self-destruct mechanism is revealed to be a jammie-dodger, he is saved by three hastily constructed spaceSpitfires
Santa Claus, of all people, gets one in Last Christmas.
The crew of Moya from Farscape get several moments. One good example was the time they attacked a Shadow Depository (Mafia bank, basically) to rescue Crichton.
Firefly: The trope name (and the quote) comes from this show, wherein Mal and the rest of the crew save Simon and River from being burned at the stake in the episode "Safe". The page photo for the trope, however, is from a different episode.
Zoe, Jayne and Wash simultaneously blasting Adelai Niska off Mal in "War Stories" is arguably another example.
Game of Thrones: In "Blackwater," the combined Tyrell-Lannister army (commanded by Ser Loras Tyrell and Lord Tywin Lannister) comes riding to the rescue of King's Landing, which was under siege by Stannis Baratheon's forces.
Happy Days: Fonz has this uncanny habit of showing up exactly when Richie or Joanie need help.
When Horatio and his party are in trouble, HMS Indefatigable saves them. It could practically be a drinking game to take a shot every time "the Bloody Indy" appears over the horizon to save the day.
However, if it is the Indefatigable that is endangered, it's Horatio to the rescue!
He saved the "Indy" who was under heavy fire from French corvettes, using a French captured ship and sailing under French colours. Not illegal in that era, but he should have changed the flag a few moments earlier to play perfectly according to the rules.
When Horatio is about to be told that he's flunked his exam, a fire ship appears and is about to destroy the "Indy". Hornblower and his examinators try to avert the event, and Hornblower and Captain Foster board the freaking fire ship — and change her course, saving the "Indy" and her men. Captain Pellew is impressed.
In "Retribution", Lieutenant Bush and crew are about to be captured by the Spaniards. Convinced that Horatio has abandoned them, Bush angrily remarks, "If you see Mr. Hornblower, tell him he'll hang from the yardarm." What happens barely half a second later? — KA-BOOM!! "Glad to see you safe, Mr. Bush!" Later in the same episode, the Spanish, now imprisoned in the Renown, stage an uprising that takes everyone off guard. Buckland is captured while Bush and Kennedy are severely wounded, and all is lost—until Horatio rams into Renown with the prize vessels and retakes the ship with his own small crews.
In "Enemy Below", both Harm and Sturgis, in their own ways, prevents Al-Qaida terrorists and a defected Russian submarine captain to unleash a dirty bomb on U.S. Carrier Battle Group in the Indian Ocean.
DI Alex Drake: I have to reclaim my destiny Arthur. Somehow, I'm going to stop you and go home.
Arthur Layton: What you talking about?
Alex Drake: I'm facing up to you. I'm strong enough. I'm strong enough to WAKE UP!
DCI Gene Hunt:(over radio) Oi, Drake! The A-Team are back in business.
Like the above example, the Big Damn Heroes moments in this series often coincide with awesome music moments too.
LOST: third season finale "Through the Looking Glass:" Three of the Others are holding Sayid, Jin, and Bernard at gunpoint. Sawyer and Juliet are hiding in the bushes, wondering how they can possibly rescue them unarmed. All of a sudden Hurley comes barreling onto the beach and runs over one of them with a Volkswagen Bus.
The best was in the series finale when Jack and Locke are having their final confrontation. All seems lost as Locke lowers his knife toward Jack's throat, then BAM! Kate puts a bullet in him.
The Man From UNCLE: Happens fairly often in the 60s spy series. Usually Solo is in James Bondage (often with the Girl of the Week beside him) when Kuryakin will swoop in at the nick of time; occasionally it happens with the roles reversed. One example: The second season episode "The King of Diamonds Affair", when Kuryakin arrives to save Solo, the innocent of the week, and a villain-turned-hero who are strapped to cannons with lit fuses which are about to fire:
Napoleon Solo: Next time, try not waiting til the last minute.
Illya Kuryakin: Next time try not to go that far up the Amazon.
Merlin: During the third season finale Gaius saves Merlin from Morgause, using his magic for the first time in years.
Mighty Morphin' Power Rangers: In the first season, this always happened concerning Sixth Ranger Tommy - Rangers would get their asses handed to them, Zordon calls in Tommy, Tommy cleans up shop. One epic moment is Tommy's I'm Back moment in "Return of the Green Ranger Part 2" when Tommy jumps in, repowered, and saves the Rangers with his Dragon Shield glowing brightly, sending the bad guy scattering. As one Youtube viewer put it, "He saved everyone with the power of his awesome!"
Without a doubt, Bulk and Skull's CMOA was during the second season during "When is a Ranger Not a Ranger", when Lord Zedd likely came the closest he ever came to true victory with a Monster of the Week who managed to strip the Rangers of their memories and their powers (all six of them). Bulk and Skull, were the ones who managed to rescue them this time; sadly, a side effect of the creatures powers caused them to forget that it happened, but that was for the best, because they also saw the Rangers unmasked.)
In Samurai Sentai Shinkenger and its American adaptation Power Rangers Samurai, the female Red Ranger (whether you want to call her Kaoru Shiba or Lauren Shiba) saves the team after the normal Red Ranger (again, either Takeru or Jayden) is injured.
Murder, She Wrote: The police force was this for Jessica, whenever the unmasked murderer attempted to add her to their body count.
Person of Interest: Typically when either Reese, Finch, Carter, or Fusco is in trouble, one of the others will come in for the save. Special mention goes for the first season finale where Fusco and Carter play this for Reese when he is pinned down by HR and soon will die.
Primeval: Nick and Cutter are being chased by a Raptor. Connor, who's been ridiculed all episode for his lack of gun skills, barrels in from behind Nick, yells him out of the way and takes a fully grown Raptor down with one shot.
In another episode, a swarm of Megopterans and a pack of future predators are catching up to the team and all seems lost...cue Becker, thought to be dead already, blasting away from his hiding place inside a car. Damn, that guy is badass.
Psych: "Big Damn Heroes" is Lassiter and Juliet's standby mode. Listing all the times they've kicked down doors with guns at the ready to rescue Shawn and Gus from the villain of the week (sometimes more than once in a single episode) would cover at least 1/3 of all episodes ever aired. The duo is usually together during these moments, although both Lassie and Jules have pulled off a handful of solo BDH's too.
Red Dwarf: Ace Rimmer (What a guy!) is usually introduced in a Big Damn Heroes scene. In his first appearance, he flies in, saves the crew, then performs emergency surgery on the Cat while nursing a broken arm. The second time, he sweeps in, rescues a Distressed Damsel from a Nazi firing squad, surfs from a plane on an alligator, takes a bullet to the holodrive and still manages to have sex before seeking aid — what a guy.
Revolution: Miles and Rachel's appearance in Annapolis - with a laser-guided missile launcher - in "The Stand." One almost expects Rachel to invoke this trope, and Miles to declare, "Ain't we just!" (from Firefly).
Roswell: In the series finale, Michael is on his way out of town when he sees a fleet of black cars heading the other way. Rightly guessing this means trouble, he turns back...just in time to burst through the school auditorium doors on a motorbike, ride up the aisle and onto the stage, where Max gets on behind him and everyone gets away.
Scrap Teacher: Takasugi and his group are always this. They show up just when their teacher and/or classmates are in some kind of trouble.
Stargate SG-1: Did the same thing as Battlestar Galactica, above, in a battle against Anubis fleet above Antarctica, in the two-parter Grand Finale (well, it was a finale at the time), "Lost City", at the end of season 7. Dozens of ships are descending towards the heroes' defenseless cargo ship. Cue rockets blowing up Goa'uld fighters, as General Hammond leads the Prometheus against the incoming enemies.
Both of those situations share several stylistic similarities: main characters convinced that they are doomed, and the sudden appearance of rockets, instantly followed by the use of a Theme Music Power-Up: Galactica with fast-paced drums and Stargate with the main theme of the series. The resemblance is probably just a coincidence, unless a Galactica writer did a Homage to SG-1.
Another example is in Stargate Atlantis at the beginning of season 2 when Atlantis is under imminent attack, the self-destruct is armed and counting...then Earth responds to their earlier distress call with a small army of marines and an intergalactic battlecruiser.
Episode "Shattered Mirror". When Mirror Bashir and Mirror Dax swoop in to save the day, you've almost forgotten that they were a part of the plan all along.
In "Sacrifice of Angels", the Defiant is leading Starfleet against Dominion-Cardassian forces. Just as three Jem'Hadar are closing in on it, the Klingons finally arrive, decloaking from the direction of a star and smashing through the Dominion lines.
But defied in the Season 4 finale, "Lucifer Rising": when Ruby prevents Dean from getting into the room in time to tell Sam that Lilith is the final seal and killing her will release Lucifer.
Done to a hideously Narm-ish degree in the Season 5 finale when Dean shows up to interrupt the impending smackdown between Michael and Lucifer. And then hilariously subverted a few minutes later when Castiel appears and hits Michael with a Holy Hand Grenade (aka, a Molotov of anti-angel holy oil) only to get blown up a second later by Lucifer.
Strange Empire: A fairly routine trope of the show, done usually by Kat Loving.
Torchwood: Ianto Jones is about to have his throat slit by a cannibal with a meat cleaver, when good ol' Captain Jack breaks through the wall "in a ruddy great tractor" and starts shooting people. Cue theme music.
Tower Prep: Ian, Suki, and CJ show up just in time to save Gabe in the season finale.
True Blood: Jason Stackhouse confronts a mob of Maryann's crazed revelers armed with only a chainsaw and a nailgun in order to save Sam from being sacrificed. Thanks to a rare moment of cleverness on his part, it works. Took a Level in Badass, indeed.
Veronica Mars: This seemed to be Logan's main function whenever Veronica got into over her head. Out of every one of them, his biggest BDH moment has to be in 2x08, when Logan points a gun at a bar full of Fitzpatricks, orders them to let Veronica go, and walks out with her. Made all the more awesome by the fact that the gun wasn't loaded.
In "Field Trip", it's AD Skinner and bunch of other FBI agents who save the dynamic duo from caves with hallucinogenic mushrooms that were devouring them alive. Both are barely conscious, covered in mud and yellow substance.