Wolves manager Mick McCarthy praised his side for battling back from a first half he described as "hopeless" and "abject" to snatch a point at Goodison Park.
Sylvan Ebanks-Blake slotted home the equaliser in the 75th minute with the visitors' first shot on target to cancel out Tim Cahill's early strike.
The goal maintained Wolves' unbeaten start to the new season and McCarthy insisted: "We got up the pitch and got in their faces and I thought we deserved a point.
"We were hopeless in the first half and that is my responsibility because I was trying to stop them playing and fiddling around with the team - none of which worked.
"They should have been out of sight at half-time. They could have been 5-0 up but at the end of the second half I'm not too sure I wanted it to end - I fancied us winning.
"I was considering making substitutions but I thought they were the best lot who started and they showed that to me. I thought we deserved our equaliser."
Such was McCarthy's bright mood he took the unusual step of issuing a public apology to referee Lee Mason for the ear-bashing he gave him after Everton's goal.
McCarthy thought Cahill had tripped Jody Craddock in the process of scoring but admitted: "I was having a bit of a go about it because I thought it was a foul.
"But I've looked at the replays and I'm not as convinced now. I apologise to the referee Lee Mason because I was having a right whinge about it."