The hosts had failed to score in four games heading into the Goodison Park fixture, but took the lead within 10 minutes as Ashley Young curled home a free-kick.
After VAR spared Wolves’ blushes for a swift Toffees second, they eventually found it through Orel Mangala to end a dismal half for the visitors.
Wolves made a dreadful start to the second half as another set piece caused chaos and Dawson turned into his own net, as the travelling fans began to turn on O’Neil.
A fourth then came, seemingly from another Dawson own-goal, as Wolves were once again battered at set pieces.
Analysis
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O’Neil made four changes from the side that lost to Bournemouth at the weekend.
Dawson, Santi Bueno and Matt Doherty all came into the back line, as Goncalo Guedes made his first Premier League start of the season in attack.
Nelson Semedo was absent through suspension.
Wolves created a good early chance as Rayan Ait-Nouri was released down the left and crossed low for Jorgen Strand Larsen, but the striker fired wide.
The visitors had started well with Andre winning everything in midfield, as Matheus Cunha had a low effort saved by Jordan Pickford.
A defensive lapse of concentration from Dawson resulted in Abdoulaye Doucoure firing over and then being flagged for offside, but referee Michael Salisbury awarded the hosts a free-kick for a foul by Santi Bueno.
Young stepped up in the 10th minute to curl the free-kick around the wall and beat Jose Sa at his near post.
Shortly after, James Tarkowski beat Dawson and Doherty in the air to head home a second, but after a long VAR check it was ruled out for offside on Mangala, who was interfering with play.
Both sets of fans chanted against the technology as Goodison Park became hostile and Wolves’ emotional side did not settle, as Dominic Calvert-Lewin was sent through but saw his effort saved by Sa.
Goncalo Guedes (Getty)
Wolves should have equalised when Doherty’s first time cross found Larsen unmarked, but he did not connect with the ball from close range.
Eventually the hosts doubled their lead as Wolves failed to clear their lines from another Dwight McNeil free-kick.
The first shot from Vitalii Mykolenko was blocked before Mangala finished from the edge of the box.
Tarkowski then rose highest from a corner but headed straight at Sa, in the latest of several warning signs for Wolves.
The teams entered the break with Wolves losing 2-0 after another defensive horror show.
As the teams emerged for the second half, Wolves made a system change to a back four as Mario Lemina pushed further forward.
But Wolves made the worst possible start as they conceded just four minutes into the half.
A fantastic corner from McNeil was right on top of Sa and as Calvert-Lewin challenged, Dawson bundled it over the line for an own goal in another show of abject defending.
Mario Lemina (Getty)
Hwang Hee-chan was introduced in the 57th minute as the travelling fans began to chant ‘you don’t know what you’re doing’ at O’Neil. That was quickly followed by chants of ‘you’re getting sacked in the morning’.
Everton then had the ball in the net for a fourth but a foul was given on Sa before Doherty hit the post for Wolves with a header at the other end.
The Toffees found a fourth from another McNeil free-kick, as Calvert-Lewin challenged for the ball and Dawson seemed to turn it home for his second own-goal.
Jack Harrison had a great chance for a fifth when the ball fell to him on the edge of the box, but he fired over. Moments later, in an almost identical situation, he blazed wide when he should have scored.
The game was mercifully over after four added minutes, as Wolves slumped to another defeat.
Key Moments
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GOAL 10 Young curls home the opening goal directly from a free-kick.
GOAL 33 Mangala smashes home Everton’s second.
GOAL 49 A Dawson own goal puts Everton firmly in control
GOAL 72 Dawson seemingly turns home a second own goal
Teams
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Wolves: Sa, Doherty (R.Gomes, 77), Lemina, Dawson, Bueno (Toti, 77), Ait-Nouri, Andre, J.Gomes (Doyle, 84), Guedes (Hwang, 57), Larsen, Cunha.
Subs not used: Johnstone, Bentley, Lima, Bellegarde, Forbs.
Everton: Pickford, Young, Tarkowski, Branthwaite, Mykolenko, Mangala (Armstrong, 89), Gueye, Doucoure (Lindstrom, 83), McNeil (Harrison, 74), Ndiaye, Calvert-Lewin (Broja, 83).
Subs not used: Virginia, O’Brien, Coleman, Patterson, Beto.