Substitute Sasa Kalajdzic flicked in a late header to give Wolves their first Premier League win of the season and leave Everton pointless and bottom of the table.
The Austria striker, who joined Wolves at the start of last season but suffered an ACL injury on his debut, leapt above Nathan Patterson to head home from Pedro Neto’s cross to get their season up and running.
After a forgetful opening period in which both sides lacked cutting edge in the final third, the contest became much more of a spectacle in the second half.
Wolves failed to find a shot on target in the first half and thought they had broken the deadlock after half-time, but Fabio Silva’s celebrations were cut short as his close-range finish was ruled out for offside.
The Portuguese striker should have converted when Everton got into a defensive muddle and presented him with a clear-cut opportunity but Silva ran through and struck wide of goal.
The two goalkeepers then exchanged stunning saves as Jordan Pickford made a sharp reflex save to keep out team-mate Jarrad Branthwaite’s inadvertent effort, while Jose Sa superbly tipped over Abdoulaye Doucoure’s stooping header.
The home faithful thought they had won it with 11 minutes remaining when Doucoure slotted in to finish off a fine move, but the assistant’s flag went up for offside and Kalajdzic struck late on to stun the hosts.
