Molineux bade farewell to the Barclays Premier League with a goalless draw as Wolves avoided an unwanted place in the record books.
One more home defeat would have made it 14 for the season and 10 in a row, equalling records both held by Sunderland, but Terry Connor's side battled hard to deny an Everton side fighting for seventh place.
The visitors certainly had the better of the chances and twice had the ball in the back of the net, but Nikica Jelavic and Denis Stracqualursi were both denied by offside flags.
Despite the scoreline, it was an open game with chances for both sides, and arguably the best of the first-half opportunities came in only the fifth minute when Marouane Fellaini's flicked header landed at the feet of Tim Cahill 10 yards out but he shot straight at Dorus De Vries.
Jelavic had scored six goals in his last four league games and he had his first sight of goal in the 14th minute when he was found by Phil Jagielka's long diagonal pass, but Richard Stearman stayed with the striker and did very well to block his shot.
Kevin Doyle and James McFadden both snatched at chances while Leon Osman wanted a penalty when he went down under a challenge from Michael Kightly, but referee Lee Mason was unmoved.
It would have been a very memorable day for Tony Hibbert had he netted his first goal for Everton after more than 300 games, and his fierce volley would certainly have tested De Vries had Stephen Ward not got his body in the way.
Wolves had their best opening in the 26th minute when Stephen Hunt's free-kick was header just over by Christophe Berra, but Everton, and Steven Pienaar in particular, were starting to take a grip on the game.
The midfielder might have done better than fire over when well placed, while his clever ball found Jelavic but he was flagged offside as he shot past De Vries in what looked a marginal decision.
The hosts were living dangerously, and Ronald Zubar would have been very grateful to see his header from Hibbert's cross drop just past the post, while at the other end an isolated Steven Fletcher blazed over the bar.
Wolves threatened right at the start of the second half as Hunt and Fletcher combined to play in Stearman on the left of the area but the ball had just too much pace on it and the defender could not control his cutback.
Hibbert had been a real thorn in Wolves' side down the right and another pinpoint cross should really have been buried by Fellaini instead of placed over the bar.
De Vries was then forced into his first real save of the match from a Jelavic free-kick 25 yards out, the keeper plunging to his left.
Boss David Moyes sent on Stracqualursi for McFadden and he was into the action immediately with a shot that was well blocked by Ward, while Sylvan Ebanks-Blake replaced Fletcher for Wolves.
The Scot, who is likely to be a target for Premier League clubs in the summer, received a very warm reception from the home fans.
There was a real lack of intensity about the game but a burst from Karl Henry provided a shooting opportunity for Kightly, who pulled his effort well wide.
Jelavic had been pretty quiet but he certainly should have tested De Vries in the 72nd minute when he beat the defence to a long ball only to pull his shot wide of the far post.
The visitors had the ball in the net for a second time moments later as Jelavic's cross was turned in by Stracqualursi at the far post but again it was ruled out for offside, although it was Fellaini, who had also lunged for the ball, who was the culprit.
Everton continued to press but the Wolves defence held firm to earn their first home point since December.