Michail Antonio guided in Gokhan Tore’s deep cross back across goal with five minutes to go, as the Cherries had to compete for the final 13 minutes with ten men after the sending off of Harry Arter for two bookable offences.
Eddie Howe handed a Premier League debut to winger Ryan Fraser and his side stood up to the atmosphere and occasion to start the contest, with neither side able to create a clear cut opening.
Steve Cook thwarted the Hammers' attack with a couple of excellent blocks but Enner Valencia did manage to test Artur Boruc as his firmly struck effort in the area was palmed away by Poland international.
Arter also received his first booking in the build-up, as he handled an attempted cross before remonstrating with the linesman – which provoked referee Craig Pawson to card the Ireland international.
After a dull first-half, where the Cherries restricted the Hammers to very few openings, the Cherries came out with real intent to start the second period - and had two golden chances to take the lead.
The first fell to Jordon Ibe who was cleverly slipped through by Arter after 51 minutes, but when in the box with the goal at his mercy, the winger elected to pass rather than pull the trigger.
Two minutes later though, it was through no fault of their own why the Cherries didn't take the lead, as Joshua King hit a venomous drive from 20-yards which Adrian did extremely well to acrobatically tip over the bar.
Adrian's save proved crucial as the Hammers went on the attack again shortly after, with Tore volleying into the side netting from close range, with half the ground celebrating as if it had gone in.
The Cherries were lively on the break though, but their attacking promise was quelled as Arter received his second booking with 13 minutes to go, as he tangled with Cheikou Kouyate on the half way line and brought down the Hammers' midfielder.
It was a clear second yellow for the Cherries man and the hosts made them pay with five minutes to go. Tore picked up the ball on the left hand side and dinked the ball to the back post, where Antonio rose highest to nod in from three-yards.
It was a bitter late moment for the Cherries who had a great late chance of their own in stoppage time. King volleyed a long throw and his sweet effort was well saved by Adrian once again, before an effort from Simon Francis produced a fantastic block by Winston Reid.
The Cherries' next opportunity to get points on the board comes against London opposition once again next week, as Howe’s side travel to Crystal Palace.
West Ham United: Adrian; Byram, Reid, Collins, Masuaku; Kouyate, Nordtveit (Fletcher 79), Noble; Tore (Obiang 89), Valencia (Calleri 63), Antonio
Subs not used: Randolph, Ogbonna, Burke, Oxford
Bookings: Byram, Reid
AFC Bournemouth: Boruc; Adam Smith, Francis, Steve Cook, Daniels; Ibe (Ake 78), Surman, Arter, Fraser (Gosling 86), Wilson (Grabban 74), King
Subs not used: Federici, B Smith, Gradel, Afobe.
Bookings: Arter, Cook
Red cards: Arter
Referee: Craig Pawson