Ki penalty earns Black Cats three points.
Sunderland secured their first win at Goodison Park since Michael Bridges’ brace helped secure a 3-1 win back in 1996.
Gus Poyet made four changes to the side that drew with Norwich City for this Boxing Day fixture with Valentin Roberge and Modibo Diakite deputizing for Wes Brown and John O’Shea, who were banned and injured respectively.
Jack Colback and Steven Fletcher also replaced Emanuele Giaccherini and Jozy Altidore, who were both on the bench.
The Toffees started the game fairly brightly, holding onto the ball but could not break Sunderland down, who had set up in a 4-1-4-1 formation when out of possession.
It was the away side who got the first half chance of the match though on a cold Boxing Day. A long ball forward was not properly dealt with by Phil Jagielka of the Everton defence, leaving Steven Fletcher to latch onto a loose back pass which Tim Howard unfortunately scrambled to safety.
Everton responded with a chance of their own as Costa Rican left back Bryan Oviedo found himself on the right flank before cutting inside and hitting a swerving shot which the well positioned Vito Mannone managed to catch with relative ease.
The game was slow to get going but burst into life after 23 minutes when Sunderland were awarded a penalty.
Midfielder Leon Osman dropped back to pick the ball up but lost his concentration and allowed Ki to rob him on the edge of the penalty area. The South Korean then had a free run on goal and had rounded Howard before the American brought him down.
Referee Lee Probert duly pointed to the spot and sent the long-serving goalkeeper off as he was last man. Summer signing Joel Robles, who replaced Osman, was the tasked with keeping out a Ki penalty as he ran onto the pitch.
In front of the Gladdis Street End, the on loan Swansea City man did not panic, slotting the ball coolly into Robles’ right hand corner and, like last season, put the Lads’ 1-0 up at Goodison Park.
Sunderland should have made it two on the stroke of half time as ten men Everton looked shaken by the goal and sending off.
Some good possession football in the Everton half saw Ondrej Celustka free on the right edge of the penalty area but the Czech full back saw his shot well blocked by Robles at his near post and Seb Larsson was unable to score the rebound from close range, with the shot stopper once again keeping the Black Cats out.
The second-half saw something of an onslaught from Everton who needed to get back in the game after the half time team talk.
Mannone was crucial to Sunderland in the second period, pulling off some magnificent saves to keep the score line at 1-0 to the Wearsiders.
Oviedo once again tried his luck from distance, but this time from further out and on his weaker foot. Ironically, it was a more powerful shot which forced a good save from the Italian, who arrived in the summer from Arsenal.
A foul on Romelu Lukaku from Roberge resulted in a free kick in a dangerous position. The set piece was to be taken by Ross Barkley, who scored a screamer of a free kick at the Liberty Stadium last week.
Almost 25 yards out, the young Englishman again delivered a good set piece and, despite hitting it on Mannone’s side of the goal, still forced the keeper into a brilliant one handed save that pushed the ball around the post and out for a corner.
The resulting corner saw Nikica Jelavic meet the ball only for on loan Liverpool ace Fabio Borini to track back on the line and clear the ball away when it looked to have been a certain goal.
It was not all one way traffic, though, as Fletcher had a very good opportunity to score late on.
Substitute Giaccherini slipped the Scot through with an inch perfect ball. The frontman, though, took a poor first touch that forced him to drag the ball wide with his right foot when he would have preferred it on his deadly left boot.
There was some late drama as Mannone was forced into a double save from a corner but the Lads managed to hold on to the vital three points.
This win means that Sunderland have kept four clean sheets in their last six league games and have managed to go four domestic matches unbeaten.