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."
He added: "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."
Everton boss David Moyes admitted his side had failed to take their chances but is refusing to press the panic button after their poor start to the season.
Moyes said: "In the first half we played well but when it's only one goal you always see it's a possibility. We controlled it in the first half but not the second.
"I thought we played fine in the first half and you probably thought the victory was only going to go one way but that's what the Premier League does to you.
"You've got to take your chances when you get them. We had some great chances to make it 2-0 in the second half. Tim Cahill played well but I don't know we gave their keeper enough to do."