Brazil forward Richarlison scored the winner with 10 minutes left at Goodison Park to hand the Toffees their second win of the Premier League season.
It was an encouraging day for Everton boss Marco Silva, who handed first Premier League starts to summer signings in Moise Kean, Alex Iwobi and Fabian Delph, the latter making his first appearance in the league since January with Manchester City.
All three made an impact with former Arsenal winger Iwobi scoring for the second successive game.
Romain Saiss and Raul Jimenez, with his eighth goal in eight starts this season, had twice drawn the visitors level.
Wolves remain without a win in four league games after suffering their first defeat of the season.
Only Bolton, bottom of League One with none, had scored fewer goals heading into this weekend than Everton.
But Silva's side ended that run within 12 minutes, which provided as many goals as in Everton's previous 180 on the pitch.
The opener was a shambles from Wolves' point of view.
Conor Coady hesitated in playing a backpass to Rui Patricio so when he eventually did, the goalkeeper was taken by surprise and could only stab the ball to Kean, who fortuitously diverted it into the path of Richarlison and he blasted into the net.
Everton, with six successive home league clean sheets having not conceded since February in their last Goodison defeat, would have been hopeful of extending that run considering their record so far this season.
But that evaporated within four minutes as Adama Traore skipped past Lucas Digne and his low cross rebounded for Saiss to fire home from close range.
Back came Everton and Gylfi Sigurdsson swung in a right-wing cross which Iwobi headed home at the far post.
Joao Moutinho, one of the Wolves quartet rested after beating Torino in the Europa League in midweek, was introduced on the hour mark.
The changes worked as Jimenez levelled 15 minutes from time, the striker stealing in at the far post behind the napping Digne to bundle home a Ryan Bennett long throw flicked on by Willy Boly.
But Richarlison had the final say, heading home a Digne cross 10 minutes from time.
Boly's second yellow card in the final seconds added to Wolves' frustration.