Harry Arter’s first Premier League goal put the Cherries ahead but a second-half strike from Dimitri Payet and a brace Enner Valencia were enough for the Hammers to take three points.
New signing Benik Afobe went straight into the starting line-up and fellow new recruits Juan Iturbe and Lewis Grabban both took places on the substitute’s bench.
The Cherries largely had the better of the first-half and had the opener their play merited after just 16 minutes.
Arter picked the ball up 25-yards out, beat his man and smashed a low effort which swerved and Adrian got a hand to it but couldn't stop the ball nestling in the bottom left corner.
Debutant Afobe then had a wonderful chance four minutes later as he met Adam Smith's cross from the right but when unmarked, he headed over the bar in the six-yard box.
But after having the better of the play, Eddie Howe's men had Artur Boruc to thank for taking that advantage into the break, as substitute Nicika Jelavic got in behind and his cross from the right took a huge deflection off Simon Francis. Boruc had to be alert to keep out the deflected effort with a magnificent stop at the near post.
Boruc was also called upon just a minute before half-time as he tipped over Payet’s swinging effort from 20-yards.
After the half-time whistle, the Hammers took that momentum from the end of the first half into the second 45.
Four minutes after the break, Matt Ritchie picked up a knock, which allowed Iturbe to come on for his first ever appearance in English football.
Iturbe’s presence couldn’t quell the Hammers pressure and they monopolised the ball as the Cherries sat back.
Howe’s side remained solid until a set-piece undid them in the 67th minute. Payet stepped up from 25-yards and curled a free-kick which took a nick off the wall, hit the underside of the bar and went in.
Seven minutes later Grabban came on for his return replacing Afobe but the Hammers had their second in the 74th minute, after Payet beat Charlie Daniels on the right-hand side and fizzed in a ball which Valencia finished from ten-yards.
Glenn Murray then came on for goalscorer Arter as the Cherries put the emphasis on getting an equaliser, but the Hammers had their third as with six minutes to go, when Valencia hit a rocket of a free-kick from just outside the left side of the area into the top corner.
That goal killed the game but Iturbe did go close to a debut goal with a curling effort from 25-yards but Adrian denied the Argentine at full stretch.
The result means the Cherries are three points above the drop zone.
Cherries: Boruc, Smith, Francis, Cook, Daniels, Stanislas, Surman, Gosling, Arter (Murray 79), Ritchie (Iturbe 49), Afobe (Grabban 71).
Unused subs: Federici, Distin, O'Kane, Pugh, Murray.
West Ham United: Adrian, Tomkins (Jenkinson 60), Ogbonna, Collins, Cresswell, Antonio, Noble, Obiang, Valencia, Payet (Song 78), Carroll (Jelavic 14).
Unused subs: Randolph, Reid, Oxford, Cullen.
Goals: Arter (16), Payet (67), Valencia (74 & 84)
Referee: Martin Atkinson
Yellow cards: Smith