Gus Poyet's men look increasingly doomed after Wes Brown's own goal condemned them to another home defeat and helped the Toffees overtake Arsenal in fourth
On a day when relegation rivals Cardiff City and Fulham both picked up victories, Brown was the unfortunate party in a largely stoic Sunderland defence at the Stadium of Light.
Sunderland looked set to keep out an Everton side who scored three against Arsenal on Sunday but the former Manchester United defender unwittingly turned home Gerard Deulofeu's cross 15 minutes from time to leave the Wearside club four points adrift at the bottom.
Roberto Martinez's charges were far from their fluent best, but got the crucial stroke of luck they needed to earn a seventh straight league win and assume pole position in the race for the fourth and final Champions League spot.
Everton now sit two points clear of the fifth-placed Arsenal, while Sunderland are deep in relegation trouble despite having two games in hand.
Sunderland had the best opportunity of the first half as Fabio Borini was denied an opener thanks to John Stones' sliding goalline clearance.
Despite Everton producing better chances in the second half, with Steven Naismith and Ross Barkley both going close, it took a Sunderland man to give them the break they needed on what was another bad day at the office for Gus Poyet's side.
A confident Everton made the early running as they dominated possession, with Deulofeu having the game's first effort on target in the ninth minute before Romelu Lukaku also stung the palms of Sunderland goalkeeper Vito Mannone.
Naismith then shot over after an excellent turn past Brown as Martinez's men began to find their feet.
However, Sunderland went closest to opening the scoring when Borini's effort was cleared off the line thanks to a sliding Stones clearance.
Borini pounced on a Leighton Baines error and rounded Everton goalkeeper Tim Howard, only for Stones to desperately keep his shot out.
The chance sparked a late spell of first-half pressure for the hosts, culminating in Adam Johnson's shot being deflected wide, and Phil Bardsley's blocked header led to a goalmouth scramble after the break.
Deulofeu's tantalising cross went unfinished at the other end before Naismith failed to shoot into an empty net after Mannone had inexplicably rushed out of his goal, only to head it into the path of the Scot.
Naismith lacked the composure to take the chance, turning over, before Barkley's shot struck the outside of the post just after the hour-mark.
Everton did get their winner though, as Deulofeu produced another of his trademark bursts into the penalty area before his cut-back struck Brown and rolled past the helpless Mannone.