Former Arsenal marksman Campbell sealed Everton's success in first-half stoppage time, after they had bounced back from falling behind to an Igor Balis penalty.
The spot-kick was the first that Albion had been awarded this season but they could not build on Balis' strike and now stand on he threshold of being relegated, just a season after being promotion.
The advantage Balis had given them from the spot lasted just seven minutes as David Weir's side equalised after 24 minutes with Campbell then grabbing the decisive strike.
Albion, who have taken just 11 points from their last 27 games, fully deserved the lead Balis gave them from the spot as they had hit the bar through a Danny Dichio header and seen Derek McInnes and Jason Koumas test goalkeeper Richard Wright before making the breakthrough.
Referee Steve Bennett had no hesitation in pointing to the spot when Everton centre-half Alan Stubbs was adjudged to have handled the ball.
Former Slovakian international Balis calmly stroked the ball past Wright to score Albion's first goal in five games.
But their advantage was to be short-lived as Everton were back on level terms when Albion were caught napping at a corner.
Mark Pembridge whipped in a right wing cross that Weir converted with a glancing header.
While Everton noticeably stepped up a gear after Weir's strike, Albion should really have regained the lead before Campbell struck.
Lee Hughes set up a golden chance for Andy Johnson with a low cross only to see the Welsh international slice wide just two yards in front of an open goal.
The full cost of Johnson's miss became clear when Everton full-back Joseph Yobo lifted in a right-wing cross.
Albion goalkeeper Russell Hoult could only deflect the ball into the path of Wayne Rooney and the England international rolled the ball across the face of goal for Campbell to convert from close range.
Everton were unable to add to their advantage with rising star Rooney denied by a smart 70th minute save from Hoult.