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.
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.
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.
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.
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.
