The Hammers made a perfect start to the match, taking a three-goal lead within 10 minutes.
Before they had even opened the scoring, the home side hit the woodwork, but they weren't made to wait long to see the net bulge, Ricardo Vaz Te grabbing two goals in eight minutes.
Kevin Nolan then made it three a couple of minutes later to all but end the game as a contest.
Rob Green made an important save from Greer shortly before the half-time interval to maintain the three-goal margin.
Vaz Te completed his hat-trick just after the hour mark with an acrobatic overhead kick before a mistake in the Seagulls' defence lead to Carlton Cole making it 5-0 just minutes later.
It was 6-0 with just over 10 minutes to go, Gary Dicker turning the ball into his own net to add insult to injury for Gus Poyet's men.
The result means that West Ham move to within three points of second-placed Southampton, while Brighton's play-off hopes have taken a blow as they drop a place to eighth.