An own goal from Lee Cattermole - which required the use of goalline technology - appeared to have given the Blues victory in Tuesday's quarter-final but substitute Fabio Borini equalised with two minutes left.
And the Italian turned provider with just two minutes of extra time remaining as Ki, who also started on the bench, stayed composed to score his first Sunderland goal and send the home fans into raptures.
Chelsea took the lead immediately after half-time when Cesar Azpilicueta's teasing cross from the right was diverted towards goal by Frank Lampard, with Sunderland's Cattermole making the decisive touch.
|VIEW FROM SUNDERLAND
|By Harry Slavin
Two super-subs left Chelsea feeling blue as Gus Poyet's Sunderland showed great courage to turn this game on its head.
The Black Cats looked down and out as the visitors threatened to run riot in the second period but the introductions of Fabio Borini and Ki Sung-Yueng really turned the tide and Jose Mourinho's men looked stunned at the prospect of extra time after the Italian levelled.
Sunderland took full advantage and Ki's goal will be one that will stay with fans here for a long time to come. It may well stick with Chelsea supporters, too, but for all the wrong reasons as Mourinho's side threw away their place in the last four.
Jose Mourinho's side peppered the Sunderland goal but were unable to find a second and were duly punished when Borini drilled home after Jozy Altidore's initial effort was blocked.
Extra time was forced when Borini wasted the opportunity to secure Sunderland's comeback but the miss did not deny their progression as Swansea City loanee Ki – who featured in the Welsh side's Capital One Cup final win last year – sent Gus Poyet's men into the last four.
Chelsea made eight changes from the side that beat Crystal Palace 2-1 on Saturday, with Lampard and Ashley Cole the headline inclusions, while Adam Johnson, Craig Gardner and Andrea Dossena were all restored to Sunderland's starting line-up.
Andre Schurrle went close in the early stages before Lampard fired over but it was a lacklustre opening period and the closest that Sunderland came was a penalty shout when Dossena's drilled cross struck the arm of Azpilicueta in the penalty area.
Chelsea burst out of the blocks in the second half, though, and took the lead in the 46th minute as Azpilicueta's ball was bundled over from close range.
Just five minutes later the visitors should have doubled their advantage as Samuel Eto'o capitalised on a wayward pass from Gardner but the Cameroon international placed his effort wide.
Lampard again went close, drilling a long-range effort just over the crossbar, and, at the other end, Cattermole's fierce strike stung the palms of Chelsea keeper Mark Schwarzer.
The visitors' second-half dominance continued with Schurrle and Demba Ba both coming close but the away side were made to rue their profligacy in the 88th minute when Altidore's effort was blocked and former Chelsea man Borini fired in the rebound through the legs of David Luiz.
There was time for Borini to score a late winner when put through one-on-one but he hesitated and was then denied by Gary Cahill, sending the match into extra time.
In the first period Luiz almost put Chelsea ahead when he volleyed Lampard's corner at Mannone, before Ondrej Celustka made a crucial clearance from Willian's teasing cross.
Schwarzer then made two superb saves, first denying Altidore's point-blank effort before producing a superb diving stop to deny Ki's header. But Ki had the last laugh, scoring late to delight the crowd at the Stadium of Light.