The result lifted West Ham one point above Manchester United in fifth.
Harry Arter scored his maiden league goal early for Bournemouth, hosting its first Premier League game under lights, and hung on to the lead until an hour in.
Dimitri Payet, making his first start in more than three months after a knee injury, equalized with a smart free kick off the underside of the crossbar, and set up Enner Valencia for the winner.
Valencia netted a second time, off another free kick; all three goals came in a span of 17 minutes.
"It was a team win but Payet made the difference," West Ham manager Slaven Bilic said. "West Ham fans love that kind of player with a bit of magic, and he has got more than a bit.
"What I admire most is his character, because that is what all true champions have got. When he makes a mistake - and I am talking about training here - he is crazy and angry at himself, and that is what makes him a leader."
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