Riku's declaration of "walking the Road to Dawn" at the end of Reverse/Rebirth. All the preceding tediousness of that story mode, not to mention the biggest That One Boss of the game (Riku Replica in Twilight Town) was almost made up for with that moment.
Axel's got TONS of these. "Now you can tell me I don't respect my elders."
Lexaeus gets a moment in Reverse/Rebirth after his last fight with Riku—both of them are exhausted, and Riku tells him he's finished and jumps to deliver the final blow. Lexaeus screams, "Don't mock me!!", and hits Riku so hard he crashes into the ceiling! This after you've already defeated him in battle! It takes Riku going dark-mode to bring him down for good.
Beast's entire script in Chain of Memories is one big Moment of Awesome.
A tad meta, but the FMVs at the start of Sora's story and at the end of Reverse/Rebirth in Chain of Memories could be considered a Moment Of Awesome for the Game Boy Advance. These are full-blown FMVs, at leastPS1 quality, with character models that could almost have been taken directly from the other Kingdom Hearts games. All this from the equivalent of a slightly more powerful hand-held SNES with poorer sound quality.
Videos do require far less rendering than 3D graphics (this is why Myst was created.)
Still, it's something other games at the time couldn't boast.
Poorer sound quality? The end credits contains the full Simple and Clean song with nearly the original quality, all from a "hand-held SNES".