Wilson nodded the hosts into the lead from a corner in the 23rd minute only for Dominic Calvert-Lewin to level it up with a header just before the break -- Everton's first away goal in the league this season.
Ryan Fraser's dangerous ball into the area flicked off Everton's Fabian Delph in the 67th minute to restore Bournemouth's lead before Wilson exploited some dreadful defending five minutes later to finish the game off.
It was Bournemouth's first home win of the season.
Everton, who could have gone third with a win, are now 11th, below Bournemouth on goal difference with both clubs having seven points from five games.
It was a bitterly disappointing afternoon for Everton who only had themselves to blame for going home empty-handed.
"It was a big frustration," boss Marco Silva said. "We lost the game and we gave some things too easily to our opponent. We have to be ruthless away from home and you have to be clinical in attack.
"Our home form is fantastic but to win our away games we have to be more aggressive and more solid so that we do not concede easy goals."
Everton began strongly in the southcoast sunshine with Calvert-Lewin denied by a brave save from Aaron Ramsdale and Richarlison unleashing a dipping pile-driver that thudded against the angle of post and bar but minutes later Bournemouth were ahead.
Everton keeper Jordan Pickford made a smart save, tipping over Dominic Solanke strike. But from a corner Everton allowed Solanke to connect with a glancing header that was then sliced up in the air by team mate Josh King and Wilson was well-placed to head in.
Richarlison was a continual danger to Bournemouth and his superb cross was headed home by a towering Calvert-Lewin in the 44th minute to restore parity.
The hosts went back in front though when substitute Fraser's dangerous delivery into the area should have been hacked clear by Delph but the former Manchester City midfielder opted to use his wrong leg and sliced past a helpless Pickford before Wilson made it three.