The first crowning moment of the game: the entire world being annihilated before the main character can even do anything. How's that for a game intro?
The Fiends' introductions, with their Badass Boast and unique mannerisms, and the fact that they are each a personification of death. Even more awesome if the player manages to beat them all.
Matador: "I swear, by my sword and capote, that I will once again prove victorious!"
Thor (yes, that Thor) knocking Isamu out in one strike and then sucking out his Magatsuhi.
One more for Thor and the player: he's the third "judge" you have to defeat to leave the Mantra Court.
Dante's jump from the Mantra Headquarters down to the main character is very interesting, considering that when you can later do the same, it leaves you with 1 HP. And in a nice case of Gameplay and Story Integration, if you recruit him and jump off, he doesn't lose any health.
When weak rich girl Chiaki fuses herself with Gozu-Tennoh, becoming a freakish version of her former self with a pulsating demonic arm. She then goes on to lead a massacre of the Manikins, who just happen to be the closest thing to ordinary humans in the game world. Watch it here.
Whenever Metatron uses Fire of Sinai. The Evil Laugh before it seals the deal.
There is one rather morally ambiguous awesome moment: after killing Metatron (that is, the Voice of God) and descending to the lowest depths of the Labyrinth of Amala to meet the Old Man in the Wheelchair in person, he infuses you with more of his power, making you a full demon and powerful enough to aid him against God Himself. All very impressive, yes, but the real moment is shown when the player character is slowly descending into the depths of the lake. As the camera passes the Old Man - as millenia of planning, waiting, and tinkering finally come to fruition - ever so slightly...the Old Man smiles.
Destroying Kagutsuchi and seeing him rage and curse, in either the Freedom or True Demon endings. In the first, despite the ferocious opposition he presents, you force your way past casting after casting of Vast Light and Infinite Light, andmakehim rebuild the world he destroyed, restoring your friends to sanity with no apparent memory of the Conception, and reviving a very grateful Yuko, who thanks you for rekindling her faith in Humanity. In the second, you blaze past the Labyrinth of Amala and reach the top of the tower... and make him commit the ultimate genocide by triggering a Class-ZApocalypse How, destroying the entirety of the Multiverse. And then comes the True Final Boss...
Heck, Kagutsuchi is one of the most awesome concepts for a Final Boss in franchise history, being a combination of the Sun, the Moon, and God/the Great Will. And on all but one route, you have to defeat it.
It's not elaborated how Hijiri managed to find out about the Conception and publish a full article that turned out to be completely right, especially without the Amala Network, but that counts as an Offscreen Moment of Awesome.