The impressive Chung-Yong Lee pulled the ball back to the edge of the box for Gary Cahill whose bouncing effort seemed harmless enough.
Green looked set for an easy catch, but inexplicably the England goalkeeper let the ball slip and Ivan Klasnic was on hand to tuck away his fourth goal in four games, giving Bolton a 2-1 lead.
Bolton had gone ahead through Lee, only for West Ham to hit back through Alessandro Diamanti.
But Klasnic’s gift from Green and a third by Cahill sealed three points for Wanderers, who moved above the visitors into 17th place.
The Hammers are now 19th and also suffered a blow when injury-plagued midfielder Kieron Dyer had to be substituted midway through the first period.