Timo Werner struck his first Blues goal 19 minutes into the contest at Tottenham Hotspur Stadium and it looked to be enough to send them into the last eight for the 14th time in the past 20 seasons.
However, Erik Lamela, who started as a makeshift striker, drilled past debutant Edouard Mendy to give Spurs a deserved equaliser with seven minutes of normal time remaining.
The first nine penalties of the ensuing shoot-out were converted before Mason Mount clipped the outside of the left post with the 10th, allowing Spurs to progress.
Chelsea made eight changes from their Premier League draw with West Brom on Saturday, handing full debuts to Ben Chilwell and new goalkeeper Mendy, while Harry Kane dropping to the bench was one of nine alterations for Spurs from their draw with Newcastle United.