The South Korea international, on loan at the Stadium of Light from Swansea, struck in the 118th minute to complete a famous victory and end Jose Mourinho's hopes of winning the trophy which gave him his first honour during his previous spell at Stamford Bridge.
Lee Cattermole's own goal looked to have sent Chelsea through to the last four until substitute Fabio Borini struck with just two minutes of normal time remaining to send the tie into extra time.
Indeed, the Italian might have won it in stoppage time had it not been for Gary Cahill's desperate challenge with the Black Cats mounting a late charge.
But Mourinho will have been fuming after the visitors had entered the closing stages of the initial 90 minutes in control and having squandered chances to end the tie.
They had taken the lead when the unfortunate Cattermole bundled Cesar Azpilicueta's 46th-minute cross past goalkeeper Vito Mannone under pressure from Frank 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 manager Gus Poyet believes is not simply down to ill fortune, although Cattermole was certainly out of luck on this occasion.
However, Borini snatched an 88th-minute leveller after Jozy Altidore's shot had been blocked to set the stage for substitute Ki.
The midfielder, a 63rd-minute replacement for Craig Gardner, had already forced a brilliant save from Mark Schwarzer, but needed not second invitation when the ball arrived at his feet with just seconds of extra time remaining.