[Swordmaster]

The Dig
I loved the whole game because of the atmosphere, but there were a few moments which really had an emotional impact on me, first and foremost the introduction which climaxes to the point where the sun emerges from behind the asteroid, the music reaches its thematical peak and Low states: "We have exactly one chance. We've got to do it right. Let all your prayers be with us". Time for shivers!

Grim Fandango
I agree with what has been said in this thread already. One of the defining moments in the game in my opinion was the cutscene between Year 1 and 2, where the camera tracks the cafeteria tower up to the sky, then dives down to the night, the cafeteria having turned into a casino. It somehow opened my eyes to see just how cinematic games can be at their best. It's sad that not many titles have gotten this far, as there's plenty of room for further improvement.