Yossi Benayoun and David N'gog got the ball rolling with goals inside the first-half hour and the Reds continued to completely dominate in front of their lowest crowd of the season and just before the hour Green was left helpless when Sotirios Kyrgiakos' header hit a post and bounced in off him.
The Reds flew out of the blocks with Argentine midfielder Maxi Rodriguez forcing Green into action at his near post in the opening five minutes.
Soon after N'gog tamely headed straight at Green after being sent clear but the Reds grabbed the lead in the 19th minute.
Steven Gerrard's free-kick found former West Ham star Benayoun, who neatly chested the ball down and watched the ball go in off the far post.
The Hammers finally came to life a minute later as Carlton Cole's shot was palmed behind by Pepe Reina.
But the game was all over as a contest in the 29th minute. Benayoun and Rodriguez combined to find N'gog and the young striker fired a fierce shot beyond Green.
Kyrgiakos and N'gog then wasted further chances to add to the lead and the busy Green turned behind a Dirk Kuyt shot just before the break as the Merseysiders continued to press.
Hammers boss Gianfranco Zola introduced Mexican striker Guillermo Franco for Junior Stanislas but it was still one-way traffic.
Gerrard thought he should have had a penalty on 55 when his free-kick smashed into Mark Noble's hand in the wall.
But they had to wait just four minutes for another goal when Kyrgiakos poked a Gerrard free-kick onto a post and the ball went in off Green's leg.
Rodriguez was denied a fourth by a Green save as West Ham waved the white flag to stay just one place above the drop zone.