
Twice Everton led - through a Martin Laursen own goal and a Joseph Yobo strike - but Gabriel Agbonlahor and John Carew both scored equalizers to clinch a point for Villa.
Everton remain favourites to take fifth on 62 points, but Villa, who have 59, with each side having two games remaining.
After a tame first half, Everton took the lead on 55 minutes.
Neat work from Yakubu sent Manuel Fernandes free on the left and after his pull-back went behind the striker, Phil Neville slammed in a shot that took a deflection off defender Laursen and went beyond Scott Carson in the Villa goal.
Martin O'Neill's side huffed and puffed but their hopes looked in vain until a three-goal burst in five minutes.
First, 10 minutes from time, Gabriel Agbonlahor finished off a Gareth Barry corner, but Everton were ahead again when Yobo volleyed home after a superb pass from Lee Carsley.
But the drama was not over as Barry again turned provider, his cross headed home by tall striker John Carew.
Earlier, Portsmouth's hopes of winning fifth spot were dealt a massive blow as they were beaten 1-0 at home by fellow European hopefuls Blackburn Rovers.
Portsmouth, with one eye seemingly on next month's FA Cup final, were never at their best and Roque Santa Cruz made them pay with the only goal of the game 16 minutes from the end after a one-two with Jason Roberts.
Portsmouth now have 57 points, two more than Blackburn.
Portsmouth manager Harry Redknapp said he did not think the league was likely to bring his side a European berth.
"For me the FA Cup was always the main route," he said. "I didn't think we deserved to lose. I couldn't see us getting beat. It was a disappointing goal defensively.
"We could have dealt with it better. I don't want to criticise any individual but it was a poor goal."
Blackburn manager Mark Hughes singled out goalscorer Santa Cruz for particular praise.
"We got a great goal with the interplay between our two frontmen and it was finished well be Roque," Hughes said.
"Roque has had an outstanding season for his first in the Premier League.
"We didn't force the issue too much in the first half and just needed to up it 10 to 15 per cent in the second half which we were able to do."
By Ben James, dpa