It was a close contest in the first half with both sides cancelling each other out at the Vitality Stadium, with the teams going into the break goalless.
However, Bournemouth started very brightly in the second period and managed to break the deadlock when in-form Wilson found the back of the net in the 53rd minute.
The striker positioned himself well when a lobbed cross found him on the edge of the area before he went on to fire home a composed strike, giving Lukasz Fabianski little chance.
West Ham offered very little in the contest with just one shot on target in the 90 minutes.
The Cherries finished the match off with an injury-time goal from Norway striker Josh King, with the Scandinavian producing a fine finish from the cross of David Brooks to seal the win.