When Gori (the monkey) kicks a headdress at the final boss of the Prehistoric story. Double points are given because that was what the evil tribal leader was wearing five seconds before Pogo, Gori, and Bel saw him get eaten.
Then the three heroes go and fight the dinosaur anyway in a Big Damn Heroes moment which is made bigger when Zaki comes to help.
Kung Fu chapter: Inheritance
The Xuan Ya Lian Shan Quan is always a Crowning Moment Of Awesome. Even when you get it as a normal technique, but especially the first time the student uses it.
By extension, anything epic the student does also counts as their former Master's moment.
A great example is when Yuan is your chosen student: When you face a big number of goons alone, he suddenly appears and starts ranting about how he sucks compared to Li and Sammo while beating effortlessly the crap out of the goons!
Made even more awesome by the gameplay: they are the weakest of the three at the beginning in every stat, but at level 16, is the best (or a close second) in every stat (except the HP, where he unfortunately stays the weakest)!
Another is the end fight to Odi Wang Lee. The student fighting him performs the technique taught last minute by the master (which you could also use once during the battle itself). And for the finishing blow, the successor makes a slow motion spin kick that sends him flying off the table, breaking through his chair, and hitting his back against the gong. Killed on the spot.
Bakumatsu chapter: Secret Orders
The hostage you are ordered to rescue can be not rescued. Technically, you must rescue him, and then he forces you to go and fight the Big Bad of the chapter; but you can skip the rescue and go to face the Big Bad alone. If you do so, he rescues you!!
The Watanabe father gets one before he dies by dodging sword strikes while in a small space. And said attacker was below him.
If you so choose, you can forgo setting any of the traps. This means you fight a dozen fully armed, dangerous men or ghosts with only a couple of guys, Spaghetti Western style - and WIN.
Present Day chapter: The Strongest
When Odie Oldbright tells Masaru how he killed all the fighters Masaru had defeated and learned from, all in cold blood, Masaru declares that Odie didn't know what a fight was and that he would use the master's powers to end Odie for his crimes.
Near Future chapter: Flow
Akira getting Buriki Daioh to move completely and then the theme song starts up. The player will then quite possibly go through a Theme Music Powerup in real life, it's that good. And then you fight the entire friggin' Japanese army!
Before that, Matsu takes a drug that amplifies his spiritual power, barges into the Buriki Daioh's cockpit, and manages, through sheer grit and force of will, to get it moving. And then he kicks a lot of ass before the overdose kills him and Akira has to take over.
Darth faces the Behemoth, the Final Fantasy monster that can one-hit-kill Cube and killed Huey and Rachel, with nothing more than some sort of gun. And wins!
When Rachel tries to kill herself and almost kills Cube in the process, Kato saves him. Also a part of the main Character Development thread of Cube progressively behaving and being considered as a human.
And of course, Cube himself by defeating the Big Bad of the chapter. Even more so considering he had no fighting ability, and hacked the virtual reality in which the fight happens to give himself some fighting abilities. That he keeps once back in the real world!
Medieval chapter: King Of Demons
The moment Oersted stopped being a Heroic Mime and proclaimed his undying hatred of the world and everyone in it. After all the horrible things that happened to him, slowly destroying everything he believed in, he finally snaps, and it is awesome.
It's probably the most emotional scene in the entire game because in theory you know that it's probably a bad idea, but you just want those asshole villagers to get their comeuppance so badly.
Pretty much the whole of Oersted's version of the Final Chapter. You get to be the villain for once as you wipe out the Heroes. His monologues at the start of the chapter and before each battle gave more of a feel to his perspective.
And what about seven people from different eras, that never met neither know, and even fight between themselves at the beginning, defeat the fucking God of Hatred!!
The overall ending itself can count as an awesome ending, simply the thought of the seven heroes running altogether on the sun as montages of their actions and the aftermath of their quests, accompanied with awesome music, is just awesome.