The only goal at the Stadium of Light came after 75 minutes when Sunderland defender Wes Brown chested a cross into his own net to give Everton a seventh successive victory and move them into the Champions League places.
They have 66 points, two more than Arsenal who were playing FA Cup holders Wigan Athletic in the FA Cup semi-final at Wembley.
Sunderland's defeat left them four points adrift at the bottom after second-from-bottom Cardiff City won 1-0 at Southampton and Fulham, who are 18th, beat fourth-from-bottom Norwich City 1-0 at Craven Cottage.
Crystal Palace eased their relegation fears with a 1-0 home win over Aston Villa but West Bromwich Albion, who started the day in 16th place, drew 3-3 with Tottenham Hotspur after leading 3-0 after 31 minutes. Spurs also missed a penalty.
Palace's victory saw them climb to 37 points and virtual safety while West Brom stayed in 16th place.
The bottom four positions are occupied by Norwich who have 32 points, followed by Fulham (30), Cardiff (29) and Sunderland(25). Sunderland have two games in hand on the teams above them.
The top three sides Liverpool, Chelsea and Manchester City are all in action on Sunday with Liverpool playing Manchester City at Anfield and Chelsea going to Swansea City.
All of Saturday's matches kicked off at 1507 local time (1407) to commemorate the Hillsborough disaster 25 years ago this weekend when 96 Liverpool fans died at their FA Cup semi-final against Nottingham Forest in Sheffield.
That match was abandoned after six minutes and the seven-minute delay on Saturday marked that duration and also included a minute's silence around the grounds.
Image: Everton players celebrate after scoring a goal
Photograph: Alex Livesey/Getty Images
EPL PHOTOS: Chelsea get the royal blues at Palace; Arsenal hold Man City
Sports Shorts: Srikanth, Sindhu in Singapore Open QF
Sports Shorts: Srikanth enters semis of Singapore Open
Sports Shorts: 'I'm scared to sleep,' tearful Pistorius tells court
Why Suarez allows Gerrard to take Liverpool penalties?