Leighton Baines's late penalty gave Everton their second league win at home this season at the expense of Wolves.
Stephen Hunt gave the visitors the lead from the spot after midfielder Marouane Fellaini was judged to have fouled David Edwards just inside the area.
Phil Jagielka equalised for Everton when he headed in Baines's free-kick from the left.
Baines secured victory from the spot after Stephen Ward was penalised for a push on Louis Saha in the box.
Everton spirit delights boss Moyes
The challenge that brought referee Jon Moss's game-changing decision in the 83rd minute appeared nothing more than a gentle nudge, but if the visitors felt aggrieved, Everton fans could point to an incident in the first half when Karl Henry's tug on Tim Cahill did not result in a penalty.
Toffees boss David Moyes will care little for the earlier decision having seen his side move away from the Premier League relegation zone and into 13th place.
And the victory may have placated some of the anger shown at the pre-match protest by supporters' group Blue Union - numbering about 250 according to Merseyside Police - calling for owner Bill Kenwright to sell the club.
DID YOU KNOW?
Leighton Baines has scored five penalties in 2011
Whatever the elation at the final whistle, there had been groans among the home support when their side went behind to a penalty shortly before half-time.
On that occasion, referee Moss's decision was not disputed, Fellaini being penalised for a trip on Edwards and Hunt calmly stepping up to score his second league goal of the season.
Seven minutes later, Everton were level through Jagielka, playing despite a broken toe.
McCarthy angry at 'soft' penalty
Baines, who had provided the home side's most reliable delivery, whipped in a free-kick from the left which was powerfully converted by his England team-mate.
Then came controversy late in the first period, when Cahill appeared to have been tugged back by Henry as the Everton man tried to follow up Fellaini's shot. The Toffees players and home fans called for a penalty, but their pleas were ignored.
The first 20 minutes of the second half was as dour as the first 20 of the opening half and Everton's fans voiced their frustrations as their team continually gave away possession.
DID YOU KNOW?
Everton have not lost to Wolves at Goodison Park since a 3-2 defeat in September 1979
When the home side did get inside the area they were first denied by keeper Wayne Hennessey, who closed down Saha, and then Ward, who produced a stunning block to prevent Cahill from scoring from two yards out.
But with seven minutes remaining, Everton broke Wolves's resistance when Baines tucked in from 12 yards after Moss had pointed to the spot.
Wolves were dealt another blow after it was revealed that defender Richard Stearman, substituted after 58 minutes, had suffered a suspected broken hand.
Everton manager David Moyes:
"The points were all-important today.
"The way we went about it and how we got there was relevant but not as relevant as getting the three points and I think we deserved it in the end.
"I thought we played well in the first half; we got some crosses in and some were good and some weren't and we got the ball in the box and didn't finish it off.
"Then before we knew it we had given away a penalty and I thought 'here we go again' but all credit to the lads for getting a goal back before half-time.
"That was really vital. We needed to find a goal from somewhere and we looked like we were huffing and puffing.
"We ended up winning the game with two set-pieces: a free-kick and a penalty."
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