I'm sort of amazed no one has nominated Fallout yet. To me it's sort of a gold standard for a gripping, well-designed intro to a game.
http://www.youtube.com/watch?v=geLiEiAiQJA I'm also giving a call-out to Sid Meier's Alpha Centauri's intro. It made good use of stock footage, voice over, and editing to paint a portrait of an earth in hyper-crisis mode, on the edge of an abyss that prompted the launch of Humanity's Last Hopein the launching of the U.N.S. Unity, where we find the game beginning:
http://www.youtube.com/watch?v=035cpHEowS4 Many of the secret project videos in that game are excellent in of themselves too. This game was one of the last examples I can think of to incorporate FMV in their design to good effect, before it pretty much wholesale died out in favor of all-CGI cutscenes.
Perhaps the most ambitious and expensive video game intro (for the cost of the salaried cast members if nothing else!) has to be Wing Commander III's mammoth 12 minute setup:
http://www.youtube.com/watch?v=njJlUL_tvlU In terms of B-movie engimatic, artsy falir, probably not. Love the use of black-and-white footage. Too bad Nocturne itself hasn't aged terribly well. Those hunks of poorly carved wood called "Animated 3D characters" are woefully to both behold and control, now.