An own goal from Lee Cattermole – that required the use of goalline technology - appeared to have given Chelsea 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.
Sunderland goalkeeper Vito Mannone scooped the ball away but the Goal Decision System – in use for the first time in the competition – confirmed it had crossed the line and an own goal was given.
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 was a member of the Welsh side's League 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 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 they 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 the lead 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 goalkeeper Mark Schwarzer.
Chelsea's 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.
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 Stadium of Light crowd.
