It was a thrilling opening 45 minutes for the people who managed to make it into a bare Stamford Bridge with César Azpilicueta going close inside the opening ten minutes.
Danny Drinkwater came even closer when he clattered the crossbar from some distance but The Canaries showed that they weren't going to be pushover by hitting the crossbar themselves soon after.
Chelsea's long wait for a goal would only last ten second-half minutes as Michy Batshuayi stabbed home the opener, the visitors looked to punish their hosts with good chances for Josh Murphy and James Maddison but to no avail but Jamal Lewis' header made sure it would go the distance with his header right at the death in additional time of normal time.
Extra-time lacked little to no opportunities with Willian going closest but the half hour belonged referee Graham Scott as he sent off both Pedro and Álvaro Morata, Nélson Oliveira's miss proved to be the deciding factor in the shoot-out as Eden Hazard set up the tie with former manager Rafael Benítez.
Can't buy a goal
The Blues headed into this replay in desperate need of a goal having pulled out only goalless draws in their last three matches including the draw at Carrow Road, Antonio Conte made a number of changes for the visit of Daniel Farke's side but despite that the same problems remained throughout the opening period.
Their first real opportunity arrived ten minutes in as Kenedy played in the corner following Willian's initial effort, the cross was missed by compatriot David Luiz but was headed towards the back post by Tiémoué Bakayoko and Azpilicueta came within inches of nudging the ball into the back of the net.
Chelsea supporters will have thought someone of a higher power was working against them as they came within a lick of paint of taking the lead, Drinkwater decided to punt one from 30 yards which was dipping and moving and Angus Gunn did well to get a touch onto it to put it onto the crossbar.
Batshuayi had been given his chance to shine by Conte but he almost gifted the lead to The Canaries, the Belgian's wayward pass was straight into the feet of Oliveira who did well to take the touch before stabbing at it with the outside of his boot which clipped the top of the upright.
Ending the frustration
As the second half began it proved to be the same old story for The Blues as they continued to struggle with breaking Farke's side down, but ten minutes after the restart Batshuayi managed to break their goal voodoo with the opening strike.
Kenedy did well to take the ball back on the left-hand side after throw-in, the Brazilian spotted Batshuayi in the middle as he rolled the ball into the near post where the striker was waiting to stab it into the top corner much to the relief of the supporters inside Stamford Bridge.
Showing they were still a danger
Despite managing to nudge themselves into the lead the visitors weren't going to keel over, as Norwich produced two very good opportunities to get themselves back into the tie.
The best of those came on the hour mark and it came from another mistake from the Premier League champions, Willy Caballero's poor punch from the free-kick landed at the feet of Murphy whose bouncing effort across goal looked destined for the bottom corner but came back off the right-hand post.
It could have easily been two for Conte's side as they quickly break from the Norwich corner through Kenedy, his through ball looked perfect for Pedro who looked to go around Gunn before hitting the turf but Graham Scott was quick to brandish the yellow card for the Spaniard for simulation.
Despite their lead Chelsea continued to hand chances to the opposition with sloppy errors, Drinkwater was easily dispossessed by Harrison Reed as he played it to Maddison who had time to set himself but Caballero did well to come off his line and save the effort.
Taking it the distance right at the death
It did look like a fourth round tie with Newcastle United all but belonged to Chelsea, but Norwich made sure that an extra half hour would be played with their effort with seconds left to play.
Timm Klose did well to get his the ball out of his feet on the left and with no pressure managed to swing a great ball into the area, defending was scarce on Lewis as he managed to glance the ball into the far corner for his first senior goal.
Trying to avoid spot kicks
It was clear that Conte wanted the match to be over as a contest as extra-time got underway introducing the likes of Hazard and Morata, however there was only one clear opportunity just before the extra-time break as Willian managed to make space but his effort was well saved by Gunn.
The tie descended into madness as the prospect of penalties drew ever closer as Chelsea's 11 became nine, having booked earlier in the clash for simulation Pedro was given his marching orders three minutes from the end as he hacked down the progressing Wes Hoolahan.
It got even worse for The Blues as they suffered their second dismissal in quick succession, Morata's run into the area was thwarted by the presence of Christoph Zimmermann but Scott had judged the Spaniard to have taken a dive for which he had received his initial booking before being sent for an early shower with his second booking seemingly for dissent.
Rounding off the marathon with a pea roller
A shoot-out awaited both sides with six yards separating them and a clash with The Magpies, and it didn't get off the best start for the away side as Oliveira's hit a sweet effort with pace but Caballero guessed right as he palmed it wide.
What followed was a great showcase of players handling the pressure of a shoot-out, but Oliveira's miss opened the opportunity for Hazard to seal the tie and the Belgian did so with ease as he rolled his spot-kick into the bottom corner.