The hosts took a while to get into their stride, struggling to string passes together as the Cherries enjoyed the majority of possession early on.
Both sides went on to enjoy bright attacking spells throughout the first half, but the Hammers were given the opportunity to take the lead from the spot after Javier Hernandez was brought down by Nathan Ake on 29 minutes.
However, the visitors were able to draw level on the hour mark when Callum Wilson picked up the ball just inside the West Ham half, ran at the home defence and, after evading several challenges slotted the ball past Lukasz Fabianski.
Eddie Howe's side then turned the game on its head by edging into the lead in the 66th minute, Ryan Fraser's free kick to the back post successfully nodded in by Steve Cook after beating marker Angelo Ogbonna.
From then on, West Ham began lacking composure and, despite a threatening late effort from Jack Wilshere, they did not create much as the south coast side held on for all three points.
WEST HAM UNITED (4-4-1-1): Fabianski; Zabaleta, Balbuena, Ogbonna, Masuaku; Snodgrass (Perez 77'), Noble (Sanchez 77'), Wilshere, Anderson; Arnautovic, Hernandez (Yarmolenko 67')
BOURNEMOUTH (4-4-2): Begovic; Smith, Cook, Ake, Daniels; Brooks (Francis 78'), Gosling, Surman, Fraser; Wilson, King (Cook 90')