Danny Graham scored his first goal for Sunderland as the Black Cats claimed a crucial win in their fight against relegation from the Premier League.
Graham hung out a leg to meet a wayward Jordi Gomez shot, the ball sneaking in off the post via the striker’s calf.
Jermain Defoe added a scrappy second as Sunderland climbed from third-bottom to 14th place for a few hours at least.
James McCarthy hit the post for the hosts who failed to score for the first time in eight league matches.
The omens were not good for Sunderland at kick-off despite their vital win over Southampton last time out.
Everton were defending an unbeaten record at Goodison Park in 2015, while the Black Cats had not won on the road since the turn of the year.
It was perhaps no surprise to see Everton boss the opening 45 minutes with Romelu Lukaku seeing an early shot blocked and Leon Osman curling just wide.
Lukaku should have done better when put clean through by Osman’s lofted pass, the striker crowded out by Costel Pantilimon.
Danny Graham’s last goal in the Premier League was scored during a loan spell at Hull in December 2013, and came against former club Swansea. Graham’s goal against Everton dropped Hull into the bottom three.
Sunderland slowly settled, with their three-man attack of Defoe, Connor Wickham and Graham providing good outlets.
But a weight of numbers up top seemed to count against them when Defoe’s firm drive cannoned off Wickham
Pantilimon twice came to Sunderland’s rescue shortly before the break, first smothering Lukaku when the Belgian again stole beyond the last defender and then standing firm to repel Osman’s low drive.
Phil Jagielka’s header deflected just wide off Wickham as Everton started the second half with a flurry of corners but the hosts were caught out by a sucker-punch on 53 minutes.
Sunderland boss Dick Advocaat had talked up how important Graham is to his side despite his failures in front of goal, and the striker repaid his faith in fortuitous fashion when he deflected Gomez’s shot past the wrong-footed Tim Howard.
Everton continued to probe without success, with McCarthy hitting the post with the best of the chances they created and Seamus Coleman seeing half-hearted appeals for a penalty turned down.
A crucial Sunderland win was sealed five minutes from the end when Adam Johnson’s drilled cross deflected off Coleman and on to Defoe, who deflected the ball in with his thigh, the ball also brushing his hand on the way into the net.
Sunderland boss Dick Advocaat: “It was good we won, it was not our best game but it was effective. It could be very important.
“Everton were the better side but it is all about winning, give 100% and show the fans what they want and that is what we did.
“They gave everything, they all want to stay up and the fans were great. They give the players something extra.”
Everton boss Roberto Martinez: “We knew it would be difficult but I thought we completely controlled the game.
“You have to score goals when you are so dominant. That gives you the game.
“Sunderland had great belief and were waiting for their opportunity. The first goal is an unfortunate deflection, the same with the other goal.”
Jermain Defoe scored his fourth goal for Sunderland since joining in January
Dick Advocaat has guided Sunderland to seven points from the last nine available
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