Ki smashed home in the 118th minute after being teed up by Fabio Borini, who had come off the bench to force extra time with an 88th-minute equaliser.
In the night's other quarter-final, Edin Dzeko scored twice as free-scoring Manchester City booked their place in the semi-finals with a 3-1 win at second-tier Leicester City.
Chelsea looked on course to reach the last four after taking the lead when goalline technology adjudged that a combination of Frank Lampard and Sunderland's Lee Cattermole had bundled a cross over the line.
But they were pegged back two minutes from time when on-loan Borini steered an expert finish past two Chelsea defenders on the line from a tight angle.
The hosts then laid siege to the Chelsea goal and were rewarded two minutes from time when Ki smashed home a low drive to send the home crowd wild.
Dzeko made his case for a more regular starting spot in City's star-studded line-up with goals either side of halftime after Aleksandar Kolarov had given the visitors an early lead.
Lloyd Dyer grabbed a consolation for Leicester.
Stoke City host Manchester United and managerless Tottenham Hotspur take on West Ham United in the last two quarter-finals on Wednesday.
(Editing by Ed Osmond)
