Lee Cattermole's own goal had looked to be sending the hosts out before Fabio Borini forced extra time, with the South Korean then sealing the Blues' quarterfinal exit.
An own goal from Lee Cattermole - which required the use of goal-line technology - appeared to have given the Blues victory in Tuesday's quarterfinal but substitute Fabio Borini equalized 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.
Chelsea took the lead immediately after halftime when Cesar Azpilicueta's teasing cross from the right was diverted toward goal by Frank Lampard, with Sunderland's Cattermole making the decisive touch.
Goalkeeper Vito Mannone scooped the ball away but the Goal Decision System - in use for the first time in the competition - confirmed that it had crossed the line and an own goal was given.
Jose Mourinho's side peppered the Sunderland goal but was unable to find a second and was 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 the side 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 lineup.
Andre Schurrle went close in the early stages before Lampard fired over but it was a lackluster 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 capitalized 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 was made to rue its 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.