DesertMan's scenario. When Sunayama's finished explaining his Evil Plan to Chaud, just as Chaud is about to hand his PET over to the villains... Lan spins around a corner, shouts "MEGAMAN, ATTACK!", and tosses his PET case, MegaMan and all, right at Sunayama's head (as you can imagine, this doubles as a Crowning Moment of Funny). In a series where almost all combat is virtual, this was very refreshing.
The PlantMan scenario, where Lan and Mega Man are kicking viral ass to save Mamoru from dying, was so Manly Tears inspiring that the whole scenario is a gigantic CMOA for the series in general!
Let's be honest, Mega Man does something selflessly heroic and suicidally stupid at the drop of a hat to the point where it stops being awesome sometime around game 2. That said, the ending of BN3 felt like they finally figured out how to do a proper Heroic Sacrifice with all the awesome due to such a moment. Even his resurrection after the credits is so well done as a CMOH that finally, FINALLY Mega has some Awesome in his pants. And then they hit us with BN4...
The fight against Serenade.EXE is also awesome. That MegaMan gets crowned King of the Undernet for winning is the cherry on top.
When the Cybertanks spread all over Electopia are taken over by Alphabugs, Lan dodges the tank's turrets to reach the jack-in port.
One of the modified tanks in the WWW base is firing bubbles at Lan that are preventing him from pressing forward. Lan can't reach the tank to jack in because of the constant bubbles being fired by the canon.  Enter Dex, who throws a boulder on the tank that plugs the bubble cannon.
On the subject of Big Damn Heroes, when Megaman gets ambushed and cornered by the rank 6, 5 and 4 Navis and they throw a boulder at him when he has no room to dodge, Gutsman appears, punches the rock into pieces and then finishes all 3 navis with a Gutshammer.
Battle Network 4
After rescuing an opponent's Navi from being offed by the mafia, their boss overhears their plan to off her. When they go to attack her, the screen goes dark for half a second and you see both the attackers on the ground. Not bad for an old lady.
What makes it more awesome is that she apparently attacks them with the Shun Goku Statsu because the sounds used for this scene are almost identical to the ones associated with the move from that series.
ShadeMan, who's completely immune to MegaMan's attacks, demands that he and Lan return the DarkChip they found. Out of options, MegaMan tosses him the chip... and then shoots it in midair, destroying it. ShadeMan is not pleased. (Fortunately, MegaMan gets a way to damage him in the ensuing battle.)
Again in the fourth game, this troper considers the ending a crowning moment of awesome. After defeating Duo.EXE, and with only an hour before a computer-controlled asteroid collides with the Earth, MegaMan tries to move the wheel controlling it, but it would not even budge. However, after the whole world starts to cheer for him, Lan goes Full Synchro with MegaMan, and the wheel that wouldn't even budge now spins so hard that it's a surprise that it doesn't fly off its axis. With the music to go along with it, it was perfect.
For those who say this game sucked, here's one moment of awesome no one can deny. Lan made some very questionable moral decisions throughtout the game, and Regal's attempts at a What the Hell, Hero? speech at the end were destroyed by the simple fact Lan still gave a damn about humanity, and Regal didn't. The game's story was thin, but it was a barely disguised morality tale that proved that those who ultimately see the good in humanity and are willing to fight for it are the best hope our race has of survival... literally.
Bonus points for being the impetus that gave Lan, Yuichirou, AND Wily the chance to ultimately prove that point by the end of MMBN5.
Battle Network 5
From MMBN5: Team Colonel, Baryl Bright Slaps Lan/Netto out of a Heroic B.S.O.D. and tells him to quit whining and kick ass. He does that, and you get to control COLONEL HIMSELF!
Team Protoman gave us Protoman's exclusive unbridled awesome, and Enzan/Chaud took a punch from Lan and turned it into a Rousing Speech that basically said "take that aggression, and do something positive with it".
In BN5, there's the Brain Navi of both games beating the living shit out of the impostors of himself and MegaMan. Cool when SearchMan does it in ProtoMan, but more awesome in Colonel, considering the fact that your Brain here is NumberMan.
Anyone forget Wily's CMOA at the end of MMBN5? He's an old man who walked into an EXPLODING VOLCANO, just to use a brilliant combination of the Reset Button and a Bright Slap to allow his own son a chance at redemption, and BOTH of them survived!
Battle Network 6
In the sixth game, MegaMan uses himself to contain a giant Internet monster, and you have to drag someone else's Navi (Heatman or Aquaman, depending on version) to find some Healing Waters to help him calm down. As that Navi begins administering the Water, the Beast begins to go berserk as it starts to take control of MegaMan. Lan and his dad are panicking. And then your substitute Navi Bright Slaps Mega back into control. Win.
Several chapters later, MegaMan gets kidnapped, and the Cybeast within him forced to manifest. He gets driven off and is tracked to the Underground - the burial place for the Cybeasts. For the last time, you have to take someone else's Navi into the area to rescue him, but unlike the previous two times, you have to fight the berserk Beast MegaMan with the Navi of your choice. This moment becomes awesome for differing reasons, depending on the Navi of choice.
Joe Mach's awesome Spanner in the Works, courtesy of beating the piss out of his former World 3 allies because his conscience couldn't stand the thought of those assholes hurting a child.
Just before the end of the sixth game, an army of Punk Navis are released into the world, and show up accordingly in the overworld in Copybots or in the cyberworld. Various NPCs react a little differently to them. In particular, a Mr. Prog in Seaside Area 1 has somehow managed to curb-stomp three of them at once. Bear in mind that Mr. Progs are not explicitly designed for combat in any way.
"THE ONLY THING YOU CAN TRUST TO THE END IS YOUR OWN POWER."
The very end of the series. MegaMan has been taken over by the Cybeast. Lan can't do anything to save him. The Megabeast then warps over to Iris and Colonel, weak from fighting the opposite Cybeast. The two unite to form the original Colonel with the emotion that became Iris. Colonel then rushes at the Megabeast, tears the Cybeast soul out of his body in a single strike and then performs a Heroic Sacrifice to destroy it once and for all.
The games themselves usually highlight moments of awesome by setting them to a rendition of each game's main theme. Special credit, however, goes to the Gamecube spinoff Network Transmission: this game doesn't use its own main theme for heroic moments, they use Kaze wo Tsukinuketenote Break Through the Wind. It's awesome charging towards the Final Boss with the triumphant refrain of the anime theme carrying you forward.
Also, from Network Transmission, if you avert doing Not Quite the Right Thing, you get to take the Professor out with style. While the Professor taunts Lan and Mega about how defeating the Life Virus R didn't ultimately stop him, his mockery dies in his throat when he realizes the Officials are on his doorstep. Turns out, Zero owed them a favor for sealing the virus he was generating, and used the leftover viruses infecting the Professor's own computer to sniff him out.
The postgame content for both the third and fifth games are each minor story arcs unto themselves, each culminating with a Final Battle against a Bonus Boss with some of the heaviest Super Weight in the entire plot. As tough as it is to get there, the payoff is entirely worth it.
No love for Dr.Wily? For starters, Wily survived being at the center of his base exploding, came back from being assimilated by Alpha AND his base exploding again to walk into an exploding volcano to scold his son and mindwipe him to start anew, walked out of that and survived his base exploding a 3rd time. Then there is his Manipulative Bastard credentials which reach their peak in 3, where he strings along Bass of all people with his silver tongue.