Rory Delap's late equaliser earned Sunderland a well-deserved point and delayed their Premiership relegation.
Rory Delap is mobbed after heading Sunderland's late equaliser
Sunderland knew a defeat could send them down and Everton went ahead after five minutes when Leon Osman turned in from close range after five minutes.
Jon Stead's first goal for a year put them level, but James McFadden scored from Phil Neville's long throw to put Everton back in front after 26 minutes.
But Delap headed home Dean Whitehead's cross with only 10 minutes left.
Sunderland started the day knowing the wrong sequence of results could send them back into the Championship and they made the worst possible start by going behind early on.
Gary Naysmith's cross found James Beattie, and when his half-hit shot was missed by Simon Davies, Osman provided a vital deflection to beat Kelvin Davis.
But Sunderland responded well and they deserved their equaliser 10 minutes later through the goal-starved Stead.
The striker had a 20-yard shot well saved by Everton keeper Richard Wright, but from Grant Leadbitter's resulting corner, the ball was nodded on by Chris Brown for Stead to score his first goal for the club.
Stead's strike visibly lifted Sunderland's spirits, but they were undone by another poor piece of defending as Everton restored their lead.
Neville's long throw was allowed to reach McFadden only six yards out, and he turned to beat Davis easily.
Everton, who had been unimpressive, almost added a third seven minutes before half-time when Tim Cahill's 20-yard drive was well saved by Davis.
And McFadden was denied by the woodwork five minutes after the break, his header bouncing to safety off the bar following Beattie's flick from a Cahill cross.
Sunderland were not playing like a side whose Premiership fate is sealed, but a lack of quality going forward meant they were unable to carve out any clear cut chances.
But they got the equaliser they deserved with nine minutes left, Delap stealing in at the far post to head Dean Whitehead's cross past Wright.
And Delap almost snatched a dramatic win for Sunderland when his rising 30-yard drive beat Wright but rebounded to safety off the upright.
Everton manager David Moyes:
"We got a good start, but overall I was never really that pleased with the way we were playing, even when we were leading at half-time.
"I thought we played a little bit better after that but if you don't get that third goal you are always going to be under pressure.
"I thought we abused some good opportunities and they came back to haunt us.
"We had great chances to put the game out of reach but didn't take them. We are frustrated because this was an opportunity to take more points."
Sunderland caretaker manager Kevin Ball:
"The players kept going and going and got their rewards. We could have had a winner because Rory Delap's strike was one of the best I've seen.
"When it hit the post I looked to the heavens and said 'come on give us a break', but to be fair there were times near the end when Everton put us under a lot of pressure.
"It was nice to celebrate a goal at last and I felt we fully deserved our point."
Everton: Wright, Hibbert, Yobo, Stubbs, Naysmith (Ferguson 86), Osman, Cahill, Neville, Davies (Arteta 62), McFadden, Beattie.
Subs Not Used: Turner, Weir, Carsley.
Goals: Osman 5, McFadden 26.
Sunderland: Davis, Hoyte, Breen, Danny Collins, McCartney, Lawrence (Le Tallec 69), Whitehead, Leadbitter, Delap, Stead (Smith 84), Brown (Daryl Murphy 75).
Subs Not Used: Joe Murphy, Neill Collins.
Goals: Stead 16, Delap 80.