Willian was shown a yellow card after he appeared to be clipped by Norwich defender Timm Klose in the penalty box, with Graham Scott, the referee, choosing not to overturn the decision, despite the availability of the VAR.
Chelsea finished with nine players after both Pedro Rodriguez and Alvaro Morata were sent off in extra time.
Antonio Conte's side looked to have booked their place in the fourth round, only for Jamal Lewis to head home a dramatic 94th-minute equaliser with the very last touch of normal time.
But how did those players rate?
Michy Batshuayi - despite scoring the only goal against the Canaries - has provided little support for the misfiring Spaniard, while Eden Hazard is also on a bit of a barren run. I like the English game as it is, with talking points.
While Pedro was guilty of a clear dive - and later sent off for a second bookable offence - the other two decisions from referee Scott, relating to Willian and Morata, provoked an angry response from Antonio Conte but there was no apparent intervention from the video assistant referee.
Scored his penalty in the shootout.
Such a game plan enabled Leicester to dominate at Stamford Bridge until Ben Chilwell's dismissal and then hold out comfortably when reduced to 10 men.
Global appearances: Lewis played three competitive games for the Under 19s in 2016 but made his Under 21 debut only last summer.
Batshuayi struggled for much of the game and a mistake by the striker almost led to a Norwich goal during a first half Chelsea largely dominated. Conte, who watched the play again on video replay, called the decision to book his forward forward for diving a "big, big mistake".
"The referee got the one right in the second half with Pedro but you have another ref looking at the Willian one on a TV screen, slowed down". The game will kick start at 19:45 GMT; 2:45 pm EST; 1:15 am IST on Wednesday, January 17, 2018. Chelsea's stand-in striker took advantage of Norwich's lapse in concentration and flicked a great finish into the roof of the net.
Hoping to dictate play from the back, the Brazilian tried to spray passes around the pitched to mixed success.
David Luiz should get a recall at the back with Andreas Christensen given a breather while Conte has suggested that youngster Ethan Ampadu will feature so he could replace Cesar Azpilicueta.