Children of an Elder God: Rei performs one in chapter 1 to save Shinji when he had been trapped by the spider-alike monster Matarael.
Tiger pulls one in Dungeon Keeper Ami. Ami was battling CrownedDeath telepathically, during the battle of Crowned Death's High Temple. Battling and losing, badly. Tiger possessed Ami while she is possessed by Crowned Death, a move so inherently suicidal that she manages to (metaphorically) sneak past the dueling minds and fill the massive, swimming fortress with high explosives using Ami's link to Crowned Death. She even briefly goes One-Winged Angel doing it, too.
Firefly: Seems like we've arrived in the nick of time. What does that make us, Thunder? Lightning Bolt: Big Mane Heroines, Flash!
In Chapter 6 of Equal And Opposite Attraction, Hayate is trapped in a pocket dimension of dark magic, slowly being mind-raped into a coma. In the next chapter, Negi hears her cry for help and acts accordingly. However it isn't so much his entrance that's flashy, but rather his EXIT.
Hail To The King: There are too many examples to count, really. The first example in the story comes when Godzilla catches a VTOL that would've crushed Shinji and Misato during Sachiel's attack, right before laying on the smack down.
With a name like Kyon: Big Damn Hero you'd expect Kyon pulls this trope often. You'd be right: saving Tsuruya from a some Yakuzatwice, Kanae from an upperclassman a few days later or to Sasaki when she was being kidnapped he can certainly do the titular trope.
Last Child of Krypton: Shinji does this constantly in the original story and in the rewrite. Some instances:
During the battle against Sachiel Shinji stopped a chunk of debris was about to crush Touji and his sister and then he proceeded to trash Sachiel right when it was about to blast Rei.
Shamshel was about to kill Asuka by crushing her head with its whips. Asuka had no weapons and was nearly out of power. Then Shinji showed up, tore a whip, bit the other one in two and killed the monster.
He also dove into Leliel to save Asuka before her robot's life support failed or she got nuked.
Just as Calvin is sentenced to life in Azkaban in The Best Seven Years, Galaxoid and Nebular rescue Calvin by freezing time and teleporting him onto their ship
This happens again in the sequel In Love and War. Calvin breaks into the Predicons base to rescue Susie. They do a pretty good job until they get to the air duct, which eats up a lot of time and ultimately causes them to get caught and captured by Waspinator of all people. And then the Maximals manage to rescue them.
Wonderful: Nearly two years before the beginning of the story, main character and super-hero Taylor performed one to save her friend Emma from several gang thugs that attacked her and her father.
XCOM: RWBY Within: Near the end of the Base Mission, Strike One arrives just in time to save Ruby from a Sectopod.
Fairy Without Wings: While the series has had a few of these moments, the really epic one that deserves a mention is the one that happens in chapte 236 with the least expectant character pulling it: Irene Belserion. She shows up unannounced, heals a dying Erza back to full strength, restores Era to its former glory, manages to injure Acnologia, and turns long running antagonist VV into a bloodstain. Certainly a far cry from her canon counterpart to be sure.
In Chapter Four of Fantasy Of Utter Ridiculousness, Alice is preparing for the unsavory task of dodging attacks from three giant fairies, whose danmaku have become massive to match. Cue the sun, literally, as Patchouli makes her entrance by dropping Sun Sign: Royal Flare on their heads.
In FateBlack Reflection, Ichigo DOES pull this off impressively. How impressively? He holds Excalibur at bay with a Getsuga Tenshou.
In And Again, Asahina gets one when she jumps in and kicks away Monobear when he tries to murder Chihiro while their strongest classmates are off gathering weapons. This is followed quickly by Mukuro, Sakura and Mondo getting back and opening fire before he can retaliate.
A later chapter has not just Sten, but also Alim Surana (the elven mage Warden) and Leliana doing something quite similar, although, in this case, they have to save him from his mind breaking down since the Archdemon's psychic attack is already over.
In The Vow, Jade knocks down the guards who are about to kill Po and the Furious Five. Then she frees the sextet and leads them to the trapped nobility of China, allowing them to free them from Lord Shen's forces.
The Day Everything Changed has Konata pulling this off by saving Kagami's life by fighting against her attacker, Sakura Takahashi, who had just given her the beatdown of her life. Despite taking more than a few brutal hits, Konata still ends up on top at the end of the battle.
First, adult Apple Bloom, Lezard, and Spike save Applejack from a group of Neo Canterlotsoldiers that have captured her.
Likewise, Fluttershy saving Rainbow Dash from another group of soldiers about to finish her off after Scootaloo left her for dead.
Shortly after, it looks like Scootaloo's forces are going to finish off the remaining protagonists, only for Rainbow Dash to save Pinkie Pie, causing their Elements of Harmony to evolve to armor, which is followed by destroying the soldiers and sending Scootaloo running.
Applejack breaks Rarity out of the dungeon and as they're escaping they're cornered by the brainwashed Princess Luna, who begins to wipe the floor with them before the other Element bearers burst through a window, and turn the situation into a successful fighting retreat.
In the Pony POV Series, The Final Battle with Fluttershy's Superpowered Evil Side, Princess Gaia/Nightmare Whisper, is going badly. The group are battered, beaten, and bloody, Rarity's pretty much completely out of it with broken legs and no energy left to use magic, and the only way to stop Nightmare Whisper seems to be Rainbow Dash performing a Mercy Kill. Thankfully, Fluttercruel fights her way out of the Jury of the Damned she was being tried by and manifests in the physical world. Not only does she give the others the motivation to summon the Elements, she fills in for Fluttershy as the Element Of Kindness, allowing them to defeat Nightmare Whisper with them.
Part 6 of Shining Armor's Arc has a moment that combines this with Gunship Rescue. When Shining and his griffon allies are about to be killed by General-Admiral Makarov's forces, the rest of Shining's squad shows up, followed shortly thereafter by the rest of the forces under his command, including several attack shuttles. Makarov wisely retreats at that point.
Directly before that, Thunderchild, one of Shining's squad, saved Shining from Makarov, who had him trapped in an Anti-Magic force field spell and was going to beat him to death.
Similarly, in Part 15, a Hooviet army is about to wipe out the village that Misfit Actual and the Anti-Hooviet Rebels are protecting, when Commander Shepherd shows up with every armed ship in Cadence's flotilla and scares them off.
The Battle of Cloudsdale in The Abundance. The abstract Loyalty has brainwashed the masses, Twilight is down for the count and just for fun he decides to drop The Doctor and Dinky to their deaths. Cue two heroic arrivals: first, Derpy Hooves performs a SONIC RAINBOOM to save her daughter and The Doctor... then Trixie arrives with Faith, an army of former prisoners, and a massive airship to turn the tide and win the battle.
Rainbow Dash pulling off the Sonic Rainboom (in this continuity, for the first time since she got her cutie mark) in order to save Esra and Tastar from a giant dragon.
Subverted in The Mixed-Up Life of Brad; when Twilight's boyfriend is being held imprisoned, she arrives and knocks out the guards. Brad imagines that any minute now she'll burst dramatically through the door and deliver a badass one-liner, but it turns out she just wants to talk to him—there's no point in freeing him anyway, as there's nowhere they can escape to. He has to persuade her to burst dramatically through the door (and deliver the one-liner).
A Growing Affection has multiple examples, but most notably: the Jonin backup teams save the Naruto Rescue Squad from the One and Three-Tails. Team Kakashi saves the original Ino-Shika-Cho from the Paths of Pein.
Drifting has plenty of these moments. They usually result in Naruto earning friends or having another girl fall in love with him.
Two of these happen in Kitsune no Ken: Fist of the Fox. Sasuke and Kushina lead a group of Naruto's schoolmates to rescue Naruto when he's taken captive by Mizuki's gang and Akatsuki to be tortured. In a later arc, Sai comes just in time to rescue Ino from being raped by Aoi.
In Catch Your Breath, Minato's Teleport Spam powers allow him to easily act as this. While he missed his cue in the Kannabi Bridge mission, the circumstances surrounding the mission where Rin originally lost her life changed substantially. A sudden revelation, Split-Personality Takeover, and averted Face–Heel Turn later, and Minato successfully rescues his team from Spiral/Swirly Zetsu.
In Reaching for a Dream Naruto has completely exhausted himself fighting Orochimaru then killing Kisame. Itachi manages to survive though critically injured, leaving Naruto to limp away in the hopes Itachi collapses before he does. Then after Itachi dies of his injuries, Zetsu appears to capture Naruto. Fortunately, Jiraiya arrived and Zetsu decided grabbing the last remaining pair of Mangekyou Sharingan was more urgent than grabbing Naruto.
A Crown of Stars: In an early chapter Asuka and Daniel travel back in time to rescue or save several people, including Asuka's mother. Asuka shot the death squad was going to execute Kyoko and then retrieved her mother before returning to the present.
Advice and Trust: In chapter 7 Rei was fighting Zeruel, alone. And she was losing badly. The Angel was about to blast her in pieces when Shinji and Asuka came along riding Unit 03 and knocked it to the ground before saying simultaneously: "Get away from my friend."
The final encounter with Kaworu Nagisa plays out much differently, thanks to this. While Rei, in Unit-00, is watching from above with a sniper rifle, Toji Suzuhara in Unit-03 leaps down and flattens Unit-02, taking it out of Kaworu's control and forcing him to face Shinji alone.
The arrival of Kaoru and Kaji in the restored Unit-04 helps change the course of Asuka's infamous last stand against the MP Evas.
Once More with Feeling: In chapter 17 Asuka fought Sandalphon inside a volcano and killed it, but she was sinking to her death... until Rei leapt inside the ocean of burning magma, dove towards her and rescued her.
The One I Love Is: Shinji performs two: the first in chapter 7 to save Rei and Asuka from Zeruel and the second in chapter 12 to save Asuka from the MP-Evas. He arrived too late to save Unit-02 but right on time to save Asuka's life.
The Second Try: Touji comes along during the battle against the MP-Evas piloting the Jet Alone to help Shinji and Asuka. Unfortunately he was unable to fight effectively.
In The Sun Soul, there are more than a few moments when the heroes swoop in to save the day at the last possible second. Although the author seems to like making thempay for their constant heroics.
In Pokémon Mystery Dungeon: Reflecting Balance, the main characters end up walking unprepared into a Monster House in Chapter 14 while looking for the Flygon who can help them find one of the missing crystals that is hidden somewhere in the desert. Luckily, the other two rescue teams that came along to help with the search show up, and distract the wild Pokémon while the main characters look for the exit to the dungeon.
In Chapter 19, the main characters end up in a Monster House again, and while better prepared this time, still find themselves in a difficult spot. Luckily, their friend Azure, who had recently evolved into a Marill, shows up and helps them fight off the wild Pokémon.
''Rebirth shows Ethan saving Whitney from Team Rocket in the nick of time.
In the fourth book of the  series, Harry and Jasmine are down for the count. Voldemort has captured them, Jasmine has been beaten and restrained, and Harry's right hand has been cut off and he is suffer blood loss. Cue Barty Crouch Jr. (a hero in this series) revealing himself in the faceless mob of Nocturne Grunts and using a teleporter to bring in the entire Hogwarts Staff, Remus, Sirius, Tonks, Arthur and Molly Weasley, Hermione's parents, Neville's grandma, Xeno Lovegood, every headmaster from the tournament, a battalion of Kalos and Johto police officers, Jasmine's parents and her uncle, and Cynthia and Diantha Malfoy. Each of whom swears to protect Harry and Jasmine with their lives.
Sailor Moon: Legends of Lightstorm: Right as Zoisite is about to kill a captured and beaten Lightstorm, Sailor Jupiter transforms without a Transformation Trinket and gives the bad guys a truly epic beatdown. The battle continues to escalate as more and more Sailor Scouts arrive on the scene to kick ass.
The Wrong Reflection, in a chapter actually titled "Big Damn Heroes", has the USS Bajor arrive at a skirmish between Terran Empire and Klingon-Cardassian Alliance patrol forces and announce its allegiance (the Alliance thought it was another Terran ship) by smashing up a Terran Sato-class* the mirror universe version of the Federation Starfleet's Excalibur-class cruiser. Also an example of The Cavalry, since the scene is partly written from a Cardassian officer's perspective.
"Frostbite" uses it twice, once offscreen when the Bajor has to go rescue a freighter from the True Way, and later when it curbstomps Ruul's ship and deploys an assault unit, led by Captain Kanril, to rescue the Breen-captured away team.
Clash of the Elements: Gemini, Geno, and Alpha pull off one of these moments in the very chapters that they end up joining the group in, saving the heroes from the outer guardian of the Lightning Palace, Joe Dark and The Dark Crystal King respectively.
Rosalina does this as well in Chapter 46, saving Mario from dying at the hands of Joe Dark
In return, Mario does this for Alex in Chapter 54 by stopping and destroying Joe Dark's sword from killing him in one-blow
Respawn of the Dead gives us Scout rushing in whacking the traitorous RED Spy over the head with his bat, when the latter had the Engineer and Sniper at both gun and knifepoint, and had just before been suggesting that he and the Engineer gang-rape Sniper.
The cliffhanger that was famously left unresolved in Hergé's rough draft of Tintin and Alph-Art, whereby Tintin is marched off by one of the villain's mooks to be covered in plastic and turned into a work of art, is resolved in the version finished by fan artist Yves Rodier by having Snowy find Captain Haddock, who shows up just in time to stop the bad guys before Tintin gets completely drowned in molten plastic.
The beginning of the "Big Damn Heroes" moment in chapter 20 of Warhammer 40,000 fanfic Toy Hammer starts off with the main character running over a Daemonhost (an Eldritch Abomination that has been summoned from a Negative Space Wedgie to possess an already unlikeable monster) with a battered pickup truck. Then offloading the elite of the Warhammer 40,000 universe's armies (who have been miniaturized).
Semi-subverted in that he still gets his ass handed to him about five minutes later, giving way to a secondBig Damn Heroes moment in the same chapter by a 10 year old girl (who has the unmodifiable psychic 'fingerprint' of The God Emperor of Mankind.).
Tales of the Emperasque has several moments when the Emperor saves the day, most notably when rescues Eldrad from being eaten by Slaanesh and proceeds to take Isha from Nurgle's captivity, starting with: