City broke the deadlock in the 29th minute thanks to a goal from Yaya Toure but had to wait until the 73rd before putting the game to bed when Toure's shot struck the post before hitting Green and nestlising into the back of the net.
Adam Johnson made it 3-0 before James Tomkins grabbed a late consolation goal for the Hammers with a powerfull header.
Meanwhile, Alan Pardew's reign as Newcastle manager got off to the perfect start as the Magpies beat Liverpool 3-1 in the late kick off at St James' Park.
Kevein Nolan put Newcastle in front after 15 minutes before Dirk Kuyt's deflected shot levelled matters just after the break. However goals in the last 10 minutes from Joey Barton and Andy Carroll wrapped up the win.
Elsewhere Blackpool continued their fine season with a 1-0 win away at Stoke with DJ Campbell getting the only goal in an entertaining game.
Aston Villa eased their relegation fears with a 2-1 win over Midlands rivals West Bromwich Albion at Villa Park.
Stewart Downing and Emile Heskey put Villa 2-0 in front before Paul Scharner pulled one back for the visitors at the death to set up a tense finale.
Everton and Wigan both remain in relegation trouble after fighting out a 0-0 draw at Goodison Park and so do Fulham who also had to settle for a goalless encounter at home to Sunderland.