Before anything else, this feeling definitely comes each time you best a unique monster. Contributing factors include taking down enemies that're sometimes stronger than bosses, masterful play to be able to defeat said monsters, and this awesome track playing in the backdrop to get one motivated to kill said monster.
Dunban during the Mechon Attack on Colony 9 before his body couldn't handle the Monado
The party's encounter with Metal Face in Colony 6 has a big one for Shulk. Metal Face is going on and on about how easily he killed Fiora, rubbing it in the party's faces. Cue the only instance of Unfinished Battle in the game as Shulk runs in to shut him up. He even lands a few scratches on Metal Face without the Monado upgrade, just because he's that pissed off.
The Wham Episode in Prison Island. Shulk prepares to deliver the Finishing Move to Metal Face, when the Silver Faced Mechon tries to stop him. It receives the blow instead, which opens its chest area wide open. Then everyone sees something that was inside of it... What seems to be a female homs, but with a mechanical body. But it doesn't stop there. The homs takes off the mask that was covering her face. We see Shulk's reaction. He's in shock: The homs piloting the Silver Faced Mechon is none other that Fiora! The party (and the player) just can't believe it. And Awesomeness ensues.
Just before that at Prison Island Zanza unlocks the Monado so it can finally work against Faced Mechon. The music changes to the Unique Monster theme as Shulk (and the player) get ready to give Metal Face his long-deserved ass-kicking.
Melia proves once and for all that she's not The Chick in the fight against Metal Face on the ice shelf. Mumkhar has the Monado and is waving it round like a new toy, he's remotely controlling Metal Face to tear out Fiora's throat if anyone moves, and the party is more or less frozen by indecision. Cue Melia, who steps forward and unleashes her ultimate attack with no hesitation whatsoever. It knocks the Monado out of Mumkhar's hand, sends him sprawling and takes out Metal Face in a single blow, leaving everyone free to kick Mumkhar's ass (which is itself pretty awesome).
Otharon blasting Xord into the ether river with a point-blank shot to the head. And immediately after that, Reyn pulling off an epic save and changing Shulk's vision when he prevents Otharon from dying.
At one point, Shulk comes dangerously close to Jumping Off the Slippery Slope when he is about to give into Zanza's goading and finish off Egil. But then he snaps out of it and chooses to spare him instead, giving a rare heroic version of a Not So Different speech about how he and Egil share the same pain of losing loved ones, and states that even if he doesn't want to kill Egil anymore, he can and will keep blocking his attacks against Bionis until they understand one another. The war would have ended right then and there, if only Dickson and Zanza weren't there to screw everything up five minutes later.
Near the end of the game, Shulk suddenly regaining his senses and racing to the rescue, leaping onto a Telethia and cutting it in half.
Dunban squaring off against Mumkhar in the snowfield after a boss battle, demanding that he tell him why he betrayed them and did all those horrible things.
Mumkhar: "I wanted to get the Monado so I could be the new hero!" Dunban: "Is that it? That's the puerile reason you killed all those innocent people?!" *PUNCH* "Ravaged our home!" Mumkhar: "I love this body! Hahaha! The power!"
The two then have a brief Single-Stroke Battle which Dunban, of course, wins, wounding Mumkhar's arm.
Mumkhar: "Dunban, you'll never win! Get on your knees and WORSHIP ME!" Dunban: "Never. I will strike you down and all that you've become, you wretched fool. The Monado will stop you! And then you will know the shame of all that you have done, traitor!"
Right before the final boss you get a cutscene where a Face Mechon comes out of nowhere to save Juju from a Telethia.
Bonus points because the Face Mechon not only looks identical to Xord, it even sounds like Xord. In essence, the same Mechon that brought ruin to Colony 6 is now protecting it and its inhabitants.
At the end of the very late-game boss fight with Disciple Lorithea, Kalien rebels, and begins to destroy her completely. Melia has some choice words for her.
The game's penultimate fight against Disciple Dickson. When you defeat his first phase, the cutscene implies he's been smacking the party around effortlessly, Shulk and the others barely holding him back, and he prepares to finish them off...until the whole party give Shulk and Fiora a Rousing Speech about how You Are Better Than You Think You Are. The two of them literally begin glowing with The Power of Friendship, to the point where even Dickson is caught off guard and doesn't know how to react. And that's when "You Will Know Our Names" kicks in, letting you know that it is ON.
The final boss fight counts as this, but specifically: Shulk being able to have visions despite Zanza's interference. 'The future doesn't belong to you!' finally makes full sense as a quote.
And a meta moment for the men and women of Operation Rainfall. They fought to get this game localized, and succeeded. Taking into account the game got the honor of getting a playable character in Super Smash Bros. 3DS/Wii U via Shulk, it's not a stretch to assume the two events are related.