After Harry Arter had been sent off for two bookable offences, Michail Antonio rose highest at the back post to head home Gokhan Tore's cross as the Hammers ended a disappointing week on a high note.
Chances were few and far between during a first half which was played at a pedestrian pace, with a left-footed effort from Tore that just cleared the bar the best opportunity of the opening quarter.
Despite starting brightly, Bournemouth offered nothing in the final third and they had to rely on Artur Boruc to keep them on level terms when he remained alert to punch clear a near-post effort from Enner Valencia that seemed destined for the net.
After a dressing down from manager Eddie Howe, Bournemouth returned on the front foot and Adam Smith soon got a solid effort on target after advancing from his full-back position.
Moments later, Jordon Ibe should have done better inside the penalty area before Joshua King drilled a rasping effort towards goal from 25 yards, only to see Adrian tip the ball over the crossbar.
Back up the other end, Gokhan Tore volleyed into the side-netting after being found by Antonio before Ibe saw Adrian comfortably deal with a low strike from 20 yards.
Bournemouth appeared comfortable but with 13 minutes remaining, Harry Arter was dismissed for a second bookable offence after pulling back Cheikhou Kouyate and it only encouraged West Ham to throw men forward during the closing stages.
Slaven Bilic introduced Ashley Fletcher as he opted to use an extra forward, and the gamble paid off as Tore produced the perfect cross from the left flank for Antonio to head home from five yards.
Bournemouth almost netted a late equaliser through King, who saw a volley punched away by Adrian, but the Hammers held on to record their first Premier League win of the season.