Wolves looked lively in the opening minutes, as Jørgen Strand Larsen fluffed an opening at goal before Matheus Cunha forced Jordan Pickford into a fine low stop.
However, the away side are notoriously slow starters and they were behind inside the opening 10 minutes, as Ashley Young notched his 50th PL goal, curling a free-kick around the wall and low into the bottom corner of Jose Sa’s goal.
That was the 16th first-half goal the West Midlands side had conceded this season, more than any other PL team, and it was a statistic that would get worse before HT.
Everton captain James Tarkowski thought he had made it 2-0 when he beat three Wolves defenders to head home from a free-kick, only for a lengthy VAR check to rule that Orel Mangala was in an offside position and affected play by blocking Mario Lemina.
The away side didn’t take advantage of that dubious VAR decision, as Larsen could not decide whether to volley or head the ball goalwards when latching onto a fierce volleyed cross from Matt Doherty, and the Norwegian striker’s tame effort glanced off his thigh and out for a goal kick.
Just two minutes later, things were looking ominous for Wolves, as Mangala latched onto the ball after Dwight McNeil’s shot was blocked, and the Belgian’s strike was deflected beyond Sá.
Coming into the game as the league’s joint-lowest scorers, the Goodison faithful were in unknown territory as their team revelled in the misery of the men from Molineux in the second half.
Immediately after the HT break, Dominic Calvert-Lewin challenged Sa and Craig Dawson to get on the end of a deep corner, with the ball eventually coming off Dawson and ricocheting into the back of the net.
Lightning would strike Dawson twice, as with 20 minutes to go he once again went head-to-head with Calvert-Lewin to get on the end of a cross, and again diverted the ball beyond his own goalkeeper.
It was a horrible night for the away side and may be the final act in the tenure of their head coach, as their leaky defence was comparable to an equally ineffective midfield and attack.
As Wolves floundered, for Sean Dyche it was a much-needed win, and in a manner not normally becoming of his goal-shy team.
They almost certainly won’t hit their city rivals for four at the weekend, but they can now go into the derby with some breathing space between them and the relegation zone.