The Wearsiders paid the price for sitting back more and more as the game wore on, and eventually succumbed in the 85th minute when Marouane Fellaini rifled a shot home from inside the 18-yard box.
Just seconds earlier the Merseysiders had an appeal for a penalty waved away by referee Martin Atkinson as Lorik Cana seemed to handle in the box, but the match official ruled the ball had hit him on the shoulder.
No doubt, David Moyes' men felt that justice had been done when Fellaini fired home the leveller.
Before the late goal the game was well and truly in the balance. For all of their sustained pressure, Everton never really tested Marton Fulop in the Sunderland goal.
Up until Fellaini's equaliser it could have been one of those days for the Toffees, as Louis Saha fired wide of the mark on three separate occasions when well-placed.
However, you got the feeling that it was only matter of time before they broke their duck for the afternoon, with the Wearsiders struggling to get a foothold on the game after half-time.
Before the break it was a far more even contest. It was the hosts who almost drew first blood with just five minutes gone.
Andy Reid swung in a corner, which was met by Darren Bent, but the striker couldn't get his effort on target.
Three minutes later Everton went closer still, Leon Osman's shot shaving the outside of the post before going wide.
However, it was Sunderland who made the breakthrough after a superb piece of attacking play. A long punt out of defence was headed wide by Bent to strike partner Kenwyne Jones.
Switching to the temporary role of right winger, Jones curled in a superb cross and Bent headed into the bottom corner with 17 minutes played.
The Black Cats could have doubled their lead three minutes later. Jordan Henderson got lucky with the bounce and raced to the byline before pulling the ball back for Bent.
The England striker flicked the ball goalwards, but Lucas Neill cleared on the line and, just 60 seconds later, the Wearsiders went close again, as Jones whipped in another great ball from the right.
This time he found the head of Reid, but the Irishman's attempt to cushion a header back towards goal came to nothing.
If the Black Cats went in with their tails up at the break, it was the Toffees who came out fighting in the second half.
Everton keeper Tim Howard made one great save to deny Paulo Da Silva but, apart from the that, the Blues goal was rarely threatened for much of the second half.
In the end, Everton may feel like they could have snatched a win at the death, but both sides, who were so desperate for a win, had to settle for a share of the spoils.