The England international took advantage of some sloppy defending from the visitors to head home the opener from close range, but Everton levelled before half-time through Dominic Calvert-Lewin.
The Toffees have spent big again this summer in a bid to challenge the established ‘top six’ for a European place, but will need to improve on the road to do so.
Ryan Fraser’s driven free-kick restored Bournemouth’s lead 23 minutes from time via a deflection off Fabian Delph before Wilson’s cool finish to lob international teammate Jordan Pickford made the game safe.
Everton have now picked up just one point from three away games this season after also failing to win at Crystal Palace and Aston Villa.
Victory takes Bournemouth above their opponents on goal difference with both sides on seven points from their opening five games.