Richarlison scored two in a manic game at Goodison Park as Everton twice recovered from being pegged back by Wolves.
The Brazil international scored the first and last goals of an odd affair which lurched between moments of quality, to a long spell of mediocrity and back again to provide an entertaining finish.
Everton handed full Premier League debuts to a trio of 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 Iwobi, scoring for the second successive game, and Delph, holding things together in midfield, particularly impressive - although Richarlison stole the show with his double.
Wolves in some respect fell victim to accommodating their Europa League qualification in midweek as Nuno Espirito Santo made four changes, including key personnel like Joao Moutino and Diogo Jota.
Romain Saiss and Raul Jimenez, with his eighth goal in eight starts this season, twice pulled Wolves level but defeat left them still looking for their first win after three draws.
Only Bolton, bottom of League One with none, had scored fewer goals heading into this weekend than Everton and their previous three league matches had produced just three goals (one scored, two conceded) so expectation was on the low side.
But that was flipped on its head within 12 minutes, which provided as many goals as in the Toffees' previous 180 on the pitch.
Two of the three goals were down to poor defending, with the opener shambolic 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 beat Coady on the line.
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.
Adama Traore, the right wing-back, skipped past Lucas Digne and his low cross rebounded invitingly for Saiss to fire home from close range - the first goal against the hosts in 549 minutes of Premier League football.
Back came Everton and Gylfi Sigurdsson, who could only have been marginally onside, swung in a right-wing cross which Iwobi headed home at the far post.
Still the madness continued with Patrick Cutrone, also on his full debut having joined from AC Milan in the summer, had two shots stopped by Jordan Pickford and then Michael Keane.
But that was where the action ended as both sides' football deteriorated into an exchange of mistakes, misplaced passes and general scrappiness.
After the break Richarlison and Iwobi continued to threaten down the flanks without producing any genuine end product but it was enough to have Wolves backpedalling and hacking clear.
Midfielder Moutinho, one of the quartet rested after beating Torino, was introduced on the hour as Nuno looked for an experienced head to get a grip on the game.
But it was his go-to guy Jimenez who came up with the goods 15 minutes from time, stealing in at the far post behind the napping Digne to bundle home a Ryan Bennett long throw flicked on by Willy Boly.
The Wolves striker took Digne's boot to the face in the process but it was to prove in vain as Richarlison had the final say, heading home a deep Digne cross 10 minutes from time.
Boly's second yellow card in the final seconds only compounded Wolves' misery.