Jose Mourinho's Blues took the lead early in the second half at the Stadium of Light on Tuesday night as Lee Cattermole was adjudged to have scored an own goal, and looked on course to see the last-eight tie through to a conclusion.
But Fabio Borini came off the bench to fire home an 88th-minute equaliser from a tight angle to force extra-time, and Ki then kept his cool in the area two minutes from the end of the additional period to sensationally win the tie for the Cats, who join Manchester City in Wednesday's draw for the next round.
The result brought an end to Mourinho's impressive League Cup record - not including penalty shoot-outs, Chelsea had not been beaten in the competition under the Portuguese's management for 15 games.
Mourinho made eight changes to his team after the 2-1 Premier League home win over Crystal Palace. Mark Schwarzer, Gary Cahill, Ashley Cole, John Obi Mikel, Frank Lampard, Kevin De Bruyne, Andre Schurrle and Samuel Eto'o all started. Branislav Ivanovic was suspended; Michael Essien, John Terry, Eden Hazard and Fernando Torres dropped to the bench; while Petr Cech, Ramires and Juan Mata were all rested.
Gus Poyet made three changes to the Sunderland side that drew 0-0 at West Ham. Adam Johnson, Craig Gardner and Andrea Dossena came in, while Borini, Ki and Phil Bardsley were benched. Steven Fletcher was not involved, while Jack Colback was ill.
Both sides enjoyed periods of first-half possession, but apart from an early flurry during which Vito Mannone had to field long-range efforts from Schurrle and Willian, neither goalkeeper was called upon in earnest.
Emanuele Giaccherini might have done better when he met Sebastian Larsson's 20th-minute free-kick beyond the far post, only to scuff his effort, and he blazed high into the stand after side-stepping Mikel with a minute of the first half remaining.
Chelsea took the lead when the unfortunate Cattermole bundled Cesar Azpilicueta's excellent 46th-minute right-wing cross past Mannone under pressure from Lampard.
Referee Anthony Taylor, if he needed it, was alerted to the fact that the ball had crossed the line by the new technology in place in League Cup for the first time, with replays later confirming the Sunderland man had got the decisive touch.
The Black Cats have now scored six own goals in 11 games, a tally Poyet believes is not simply down to ill fortune, although Cattermole was certainly out of luck on this occasion.
Eto'o could have killed the tie off within minutes after he found himself in on goal with just Mannone to beat, but he steered his right-foot attempt wide of the post.
However, Borini snatched a late leveller in normal time after Jozy Altidore's shot had been blocked.
The Italian might have won it in injury time had it not been for Cahill's last-gasp challenge, but Mannone then had to save from David Luiz four minutes into extra-time.
Borini stabbed wide from Dossena's deep cross, but Essien repelled Altidore's goal-bound effort after he had helped on Ki's shot.
Schwarzer produced a superb save to keep out on-loan Swansea man Ki's 114th-minute header, but there was nothing the 41-year-old could do to deny him four minutes later when he cut inside from Borini's pass and thumped the ball into the bottom corner.
Sunderland will be joined in Wednesday night's semi-final draw by Manchester City and the winners of the two remaining last-eight ties - Stoke v Manchester United and Tottenham v West Ham. The two-legged semis will take place in January.
Chelsea must pick themselves up for Monday's crunch clash at Arsenal, while Poyet's Cats will be chasing crucial points at home to Norwich on Saturday.
|0||1st Half Goals||0|
|5||Shots on Target||5|
|6||Shots off Target||13|
|51.5||1st Half Poss.||48.5|
|38.4||2nd Half Poss.||61.6|