Chelsea led for the entirety of the second half following Lee Cattermole's own-goal after the restart but with the Blues searching for a second goal to kill off their opponents, the Black Cats took advantage with two minutes remaining.
Jozy Altidore collected the ball in the area and attempted to manufacture space for himself but his shot was blocked and fell to the on-loan Borini, who fired home from a tight angle to score his second goal of the season.
With the match in extra-time, both sides sought a winner to stave off penalties and Ki proved to be the difference when he put Sunderland ahead after 118 minutes of play.
The on-loan Swansea midfielder had forced Chelsea keeper Mark Schwarzer into a great save five minutes earlier but he would not be denied a second time when he collected Borini's pass, darted inside and thumped the ball into the bottom corner to score his first goal for the club.
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