Gus Poyet's men had picked up just three points on the road in the top flight prior to their trip to Merseyside, but Ki Sung-Yueng's first-half penalty boosted their survival hopes and inflicted a first home league defeat of the year on Everton.
Leon Osman and Tim Howard contrived to gift Ki a golden opportunity to put Sunderland ahead after 25 minutes.
Howard played the ball to Osman from a goal-kick, only for the midfielder to be caught in possession to allow Ki in and as he attempted to go around Howard the United States goalkeeper brought him down.
To make matters worse, Howard was given his marching orders while Osman was quickly replaced by substitute goalkeeper Joel Robles after the penalty had been dispatched.
That goal proved to be decisive although neither side move in the table as a result, with Poyet's men staying bottom while Everton remain fifth.
Roberto Martinez made just one change as Osman, equaling David Unsworth's record of 302 Premier League appearances for Everton, replaced Ross Barkley.
Counterpart Poyet made four alterations, with two coming in defence as Modibo Diakite and Valentin Roberge replaced the injured John O'Shea and suspended Wes Brown.
Everton left-back Bryan Oviedo had the first effort on target, creating space before firing a swerving left-footed effort at Vito Mannone in the eighth minute.
Jack Colback forced Howard into a save from outside the area seven minutes later in a frantic start.
Osman was then left red-faced when his gaffe resulted in Howard upending Ki and the Swansea loanee converted the penalty to end Sunderland's wait for a first away goal in the top flight since August.
Robles was called on to make a smart double save 10 minutes after the goal but Sunderland were unable to add to their lead before the break despite enjoying large spells of possession.
Diakite almost doubled the visitors' lead a minute after the interval, nudging an effort over the bar from close range before the hosts began to pile on the pressure.
Romelu Lukaku had Mannone sprawling to his left before a last-ditch Lee Cattermole tackle denied Barkley a shot at an open goal.
Mannone was again in fine form to keep out Barkley's free-kick before Fabio Borini headed Phil Jagielka's header off the line in the final 10 minutes as the hosts desperately sought to preserve their 10-game unbeaten record in the league.
Their pressure was to no avail however as Sunderland held on to secure their first win at Goodison Park since 1996.
