The Premier League champions appeared on course for round four thanks to Michy Batshuayi’s opener but Jamal Lewis struck in the 94th minute to send the game in to extra time.
The Blues huffed and puffed looking for a winner with frustration levels boiling over as both Pedro and Alvaro Morata were sent off before they eventually won the game in the shootout, Eden Hazard converting the winning spot.
That does not tell the whole story of another difficult night for Antonio Conte’s men, who came close to being victims of an upset against a well-organised Norwich side.
Chelsea, who made nine changes from the draw against Leicester, once again struggled to create opportunities as they sort to end a run of three consecutive goalless draws.
Cesar Azpilicueta should have put the Blues ahead after 11 minutes when Tiémoué Bakayoko flicked on a Kenedy corner to the Spanish defender, who just couldn’t steer the ball in at the far post.
Danny Drinkwater also went close when his long range effort was tipped on to the bar by Angus Gunn.
Having been on the back foot for much of the first half, Norwich were nearly gifted an opening when Batshuayi gave the ball to Nelson Oliveira on the edge of the box, the Portuguese clipping the Chelsea crossbar with his strike.
Batshuayi, on his 50th appearance for Chelsea, made up for a lacklustre first half by giving the home side the lead 10 minutes after the restart, taking advantage of some statuesque defending to tap in Kennedy’s low cross.
Norwich did not crumble having fallen behind and should have drawn level minutes after falling behind, Josh Murphy’s effort bouncing through the box and clattering in to the post.
Willy Caballero then dug out Drinkwater after he made an error leading to James Maddison having a one-on-one, only for the Chelsea stopper to make a big save.
Chelsea recovered their composure and had chances to kill off the game, Morata going close twice after coming off the bench.
The Blues would be made to pay for their lack of cutting edge when Lewis glanced Timm Klose’s header over Caballero in the final minute of stoppage time to send the match to extra-time.
Willian thought he had won his team a penalty when he was brought down by Klose inside the box. VAR was not used to check the decision and referee Graham Scott eventually booked the Brazilian for diving.
Gunn had to be sharp to deny Willian and Morata as Chelsea started to turn the screw in extra-time.
Chelsea’s job was then made harder when they had two players sent off in the final three minutes of extra time. Firstly Pedro was given a second yellow card for a clumsy tackle before Morata saw red for speaking out of turn to the referee, who had just booked the Spaniard for diving.
Norwich were the first to blink, Caballero making a great stop to keep out Oliveira’s spot kick.
That one miss proved to be enough as Hazard stroked home the winning penalty to set up a fourth round tie against Newcastle.
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