The Spanish playmaker grabbed a much-needed point for the visitors at the Stadium of Light with an 83rd-minute shot that wrong-footed Craig Gordon after the ball took a deflection off Phil Bardsley.
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Sunderland had been heading for all three points after striker Danny Welbeck scored a goal in each half in response to Australian international Tim Cahill's sixth-minute header for Everton.
Welbeck, now with three goals from two games, grabbed the equaliser in the 23rd from 10 yards and made it 2-1 in the 70th with a close-range header.
The draw left Sunderland, who stunned champions and league leaders Chelsea 3-0 at Stamford Bridge eight days ago, in seventh place.
Everton moved up to 14th, two points clear of the relegation zone.
Cahill could have doubled his tally, Michael Turner hacking the ball clear off the line before colliding with the upright in the second half and Welbeck missed out on a hat-trick when he shot wide in the dying seconds.
Everton could also have snatched a late winner when substitute Jermaine Beckford missed from 10 yards in stoppage time.
"I think a point's a fair result," Sunderland manager Steve Bruce told BBC radio. "The last 30 seconds really summed it up for both teams. We could have won it, they could have won it."
