The Legend Of Zelda: Majora's Mask when the mask salesman, skull kid and Link are talking about how every meeting leads to an ending was pretty emotional.
Even more than that, the last battle of Ocarina of Time game me chills. Having played through old Zelda games, and the constant mention that Ganon was once Ganondorf but never seeing him, it was cool to finally meet Ganondorf in Ocarina. But then, just when you think the game is over, Ganondorf pops out of the ruined temple, uses the power f the triforce to transform, and the one word appears at the bottom of the screen: "Ganon." So fucking bad ass. Everything they do makes the moment just thrilling, even down to the slight change of forgoing the "type" in the bos name that appeared throughout the game. We don't need no type information. He is, simply, Ganon.
Also, it's not a cut scene really, but the moment where Mother Brain kills the Metroid hatchling at the end of Super Metroid was pretty close to an emotional moment for me. Granted, I was 12 when it came out, but still.
My consoles and the fates they suffered:
Atari 7800 (Sold), Intellivision (Thrown out), Gameboy (Lost), Super Nintendo (Stolen), Super Nintendo (2nd copy) (Thrown out by mother), Nintendo 64 (Still own), Super Nintendo (3rd copy) (Still own), Wii (Sold)
A more detailed history appears on my profile.