Don't let the dated Playstation 1 graphics fool you, Vagrant Story was a masterpiece of narrative storytelling, setting the standard for the prettier JRPGs that were still to come. The opening of Vagrant Story was about as cinematic as you could get at the time, perfectly setting up the moody fantasy tale.
9. Red Alert 2 - Introduction
If you're in the market for cheesy cutscenes, it doesn't get better than the Command & Conquer series. The blend of dodgy CGI, hammily acted live-action and in-game footage makes it feel like you're watching an 80's B-movie, as this glorious cutscene depicting Russia's invasion of the USA demonstrates.