Gabriel Agbonlahor's late strike rescued a vital point for Aston Villa to help allay their relegation fears.
Joleon Lescott celebrates putting Everton ahead with an early header
Joleon Lescott headed Everton in front from Mikel Arteta's free-kick and they also hit the post before half-time through James Vaughan's vicious drive.
Villa were poor early on but improved after the break and John Carew forced Tim Howard into two good saves.
The Everton keeper denied Carew again before the end but this time Agbonlahor followed up to slot home the equaliser.
Villa are not safe yet but this point gives them a little more breathing space at the bottom of the table.
Martin O'Neill's side are still without a win since 3 February but they climb to 14th in the Premiership and are now five points above the relegation zone with seven games left.
The home side looked devoid of confidence in the early stages, even before Lescott put Everton ahead on 15 minutes with his first goal for the club.
The defender nodded the ball goalwards after Arteta's free-kick found its way to him through a crowded penalty area and Villa defender Wilfred Bouma could only help the ball into the net.
That rocked Villa but they were unable to respond with any conviction.
O'Neill's men were seeing plenty of the ball in midfield but their passing was atrocious and their front-pairing of Carew and Ashley Young had next to nothing to work with.
It was a different story at the other end though as the pace and power of Vaughan and Andy Johnson caused Villa's defence plenty of problems.
Vaughan came close to making it 2-0 when his 25-yard shot clipped the outside of the post with Villa keeper Thomas Sorensen beaten.
And Johnson felt he should have had a penalty before the interval when he was caught by Stilian Petrov's lunge as he ran on to Arteta's quick free-kick - but referee Howard Webb felt the Villa midfielder got enough of the ball.
Villa were grateful to go in at half-time just one goal down but whatever O'Neill said back in the dressing room obviously had an effect.
With Patrik Berger on for the ineffective Gavin McCann, Villa packed their midfield and at last began to put Everton under pressure.
Carew brought a fine save from Howard, who recovered to push the ball away from Agbonlahor, before Alan Stubbs blocked Carew's follow-up.
And the American keeper did well again soon after to stop Carew's volley after Agbonlahor headed Berger's cross into his path.
By now Villa were well on top and Everton were hanging on for dear life.
David Moyes' side dug in and it looked like they had done enough for a victory that would have put them back in the top six of the Premiership - and into the Uefa Cup places - when Agbonlahor found the net with seven minutes left.
Howard came out sharply to kick the ball away when Carew broke clear but the ball fell for Agbonlahor on the edge of the area and he fired into the unguarded net.
Aston Villa manager Martin O'Neill:
"In the first half we were very poor. We were nervous and unable to hold on to the ball.
"But in the second half, considering we could have been dead and buried, we came roaring back and we definitely got what we deserved in the end.
"The crowd galvanised us and, though I feared we might not get the goal, we did - and at least we got a point."
Everton manager David Moyes on Stilian Petrov's challenge on Andy Johnson:
"It is another one for the collection isn't it?
"It was an old-fashioned tackle from behind and if it had happened on the half-way line it would be a free-kick."
"If we had got that penalty it would have been 2-0 but in the end I think possibly a draw was a fair result."
Aston Villa: Sorensen, Bardsley, Mellberg, Cahill, Bouma, Agbonlahor, Petrov, McCann (Berger 46), Barry, Carew, Young (Maloney 41). Subs Not Used: Taylor, Davis, Hughes.
Booked: McCann, Mellberg, Agbonlahor.
Goals: Agbonlahor 83.
Everton: Howard, Hibbert, Yobo, Stubbs, Lescott, Osman, Carsley, Neville, Arteta, Johnson, Vaughan. Subs Not Used: Wright, Naysmith, Beattie, Nuno Valente, Anichebe.
Booked: Vaughan, Arteta, Lescott.
Goals: Lescott 15.
Ref: H Webb (S Yorkshire).