Kairi jumps off a balcony to help Sora and is completely unharmed by it! To hell with getting her own keyblade afterwards, THAT was her Moment of Awesome.
Well, that keyblade event was pretty nifty though....
Kairi restoring Sora to human form by hugging him when he's a Heartless. You can say it's cheesy if you want, but had that not happened, the story would be over, with Ansemnote actually Xehanort's Heartless victorious and the universe screwed. Doubles as a Heartwarming Moment.
Heartless Sora in general, once you think about it. His heart was so powerful even the power of absolute darkness couldn't suppress him. This is an achievement only the Big Bad was ever willing to previously make, showcasing just how well-suited Sora is as The Hero.
Oh come on now, no listing of Kairi examples on this page would be complete without mentioning that she wrote the "One Sky, One Destiny" poem. If that wasn't awesome enough as it were, the poem actually saves Sora and Riku from death by darkness at the end of Kingdom Hearts II, as it restores the hope in Sora's heart, which opens a doorway out of the realm of darkness into the realm of light. Cue Happy Ending.
That poem is a Moment of Awesome for Disney/Square Enix, for referencing it in the FIRST Kingdom Hearts. Seriously, bust out your PS2, stick in KH, and wait a while on the title screen. A cutscene starts playing, featuring the poem at the beginning and an orchestral version of Simple And Clean/Hikari. Same thing happens in Kingdom Hearts II.
It's also in the first Kingdom Hearts's instruction book, so it might be a sort of tag line for the whole series.
Most people think Naminé's a wussy little girl. They tend to forget that despite threats, isolation, and manipulation, she managed to stand her ground in the face of Larxene and Marluxia. Then in the second game, she slips under the radar and helps Roxas regain his memories while piecing together Sora's, eludes capture and extermination from two separate parties throughout most of the course of the game, busts Kairi (and Pluto) out of prison, and did all of the above while being the only character lacking the advanced skills of her fellow nobodies OR a keyblade.
Might not count for just being Hi-Res cutscenes, but the Another Side, Another Story: Deep Dive and Birth By Sleep secret endings were just awesome. The first showcases Roxas the Dual WieldingOne-Man Army tearing through an army of Heartless, with Oathkeeper and Oblivion in glorious Hi-Res. The second is a pretty cool swordfight, but is truly awesome because it's basically a glorified magic-fest with both sides firing spells flashier than Sora could ever hope to pull off without his Drive forms (directed Keyblade tornado anyone?) The magic spells are made that much cooler by being in Hi-Res. Both are extreme cases of Cutscene Power to the Max.
The original "Another Side Another Story" was pretty awesome as well, but the expert reward sucks. Riku was way cooler with the blindfold on and Roxas wasn't much better. The reward screen shows the cast making faces at you.
Also, Riku, er Ansem, er Xehanort's Heartless may have initiated it, but Maleficent becoming a dragon and That One Boss is both a crowning moment for her, and for Sora and co. when they finally defeat her.
She also turns into a dragon of her own will in Birth By Sleep, complete with her "All the powers of HELL!" line from the original movie.
And then there's Maleficent breaking the digital Sora's keyblade in Coded.
Which leads into a Moment of Awesome for data-Sora at the end of the area, where he gains his own keyblade. Not a data duplicate, a real keyblade. This facsimile not only gained his own heart over his travels, but it was strong enough to manifest a keyblade of its very own!
A meta example: Square combining Kingdom Hearts: Final Mix, Re:Chain of Memories, and 358/2 Days into an HD re-release. This was not just for the japanese, but for EVERYBODY ELSE! Although 358/2 Days didn't have any gameplay, fans were still happy there were still be able to experience the cutscenes in HD.
It's just happened again with the announcement of the HD 2.5 Remix, which will contain Kingdom Hearts II: Final Mix, Birth By Sleep: Final Mix, and a theater mode for Re:Coded. While Dream Drop Distance is still handheld-only for now, this means the whole story (rereleases included) is now going to be avaliable on PS3 (and 3DS for KH: 3D and the playable versions of 358/2 and Coded) world wide, finally shattering the series' infamously furiating No Export for You embargo.