The Toffees were firm favourites to extend their unbeaten league run at Goodison Park in 2013 but goalkeeper Tim Howard handed them an uphill struggle when he was sent off after just 23 minutes.
The American stopper brought down Ki Sung-Yueng after Leon Osman had miscontrolled the ball just outside the area and the Korean nipped in to steal the ball before taking it around Howard.
Ki netted the resulting penalty, with Joel Robles unable to stop Sunderland taking the lead after replacing Osman.
Everton pushed and pushed for an equaliser in the second half, with Ross Barkley and Romelu Lukaku both going close on several occasions, but they found goalkeeper Vito Mannone in inspired form and Sunderland held out.
However, the Black Cats' relegation rivals Crystal Palace and Fulham also won on their travels on Thursday. West Ham drop into the bottom three, and Sunderland are three points away from 17th-placed Palace.
Everton boss Roberto Martinez had made one change to his first XI from last weekend's 2-1 victory at Swansea. Barkley dropped to the bench, with Osman taking his place. Full-back Leighton Baines was not part of the squad, despite the talk in the build-up to the contest of it perhaps being the one in which he would make his comeback after a toe injury.
Sunderland's line-up showed four changes from their 0-0 home draw with Norwich, with the injured John O'Shea and suspended Wes Brown being replaced in defence by Modibo Diakite and Valentin Roberge, while midfielder Jack Colback and striker Steven Fletcher came in for benched duo Emanuele Giaccherini and Jozy Altidore.
Everton did not look at the races early on - Howard had to rush out to clear from in front of Fletcher after Phil Jagielka almost sold him short - and calamity struck midway through the first half.
Osman stumbled as he attempted to control a short pass from Howard, allowing Ki to seize on the ball and take it around Howard, who brought down the Korean.
Referee Lee Probert pointed to the spot and showed a red card to Howard. Joel was brought on in place of Osman for his league debut for Everton, but Ki kept his head under pressure and sent his spot-kick low to the Spaniard's right and into the corner on 25 minutes.
Joel had to save well from Seb Larsson soon afterwards after initially spilling an effort from Ondrej Celustka.
The Toffees were playing with nine men for a few minutes at the end of the half after Kevin Mirallas disappeared down the tunnel, a period in which Gareth Barry was booked, resulting in a one-match ban. Mirallas was suffering from illness, and was replaced by Barkley after the break.
Mannone parried a Jagielka header and got his hands on Lukaku's attempt at a rebound shot. Lukaku then saw the Italian get down to claim his drilled shot from outside the box, before Barkley sent another ambitious effort too high.
Everton kept pressing, with Barkley looking in increasingly confident mood as he drew a parry out of Mannone and then struck narrowly wide.
Mannone comfortably dealt with further shots from Lukaku and Barkley, then produced a delightful save to push behind a rising drive from Bryan Oviedo.
There was a brief respite for the visitors and Fletcher dragged an attempt the wrong side of the post having been put through.
But Everton were soon pressing again, and Mannone did superbly to palm away Barkley's free-kick.
The Toffees came close again at the death, with Fabio Borini clearing an effort from substitute Nikica Jelavic off the line, before the final whistle blew.
Sunderland are back in action at Cardiff in a live Saturday Night Football encounter, while Everton - who remain fifth after only a second defeat of the league season - entertain Southampton the following day.