It was the perfect start with Robert Kenedy giving the home side the lead after 40 seconds, but it was Alex Neil's side who managed to get themselves back into the match with half chances from Cameron Jerome and Wes Hoolahan, before the killer blow was controversially added late in the half from Diego Costa.
The Canaries started the second half brightly with Jerome missing a spectacular chance just after the restart before Nathan Redmond got one back for the home side 22 minutes from the end. Gary O'Neil, Nemanja Matic and Russell Martin came close towards the end but the Blues held out for the three points.
Straight out the traps
Half of Carrow Road hadn't got to their seats before Guus Hiddink's side took the lead inside the first minute.
It was a simple move as Eden Hazard played it down the left to Kenedy, who showed great trickery to cut inside before taking it back onto his left foot and hitting a low shot into the bottom corner to silence the arriving crowd.
Despite falling behind so early on, Norwich dusted themselves down and got into the match and made some good chances to get a equaliser.
Hoolahan played a hopeful ball in from the left which found Jerome, but the striker could only head it over the crossbar. Jerome then turned provider eight minutes later when he tried to get a cross in from the byline but Thibaut Courtois came off his line to easily collect. Hoolahan tried a similar ball to that of nine minutes previous but Branislav Ivanovic was equal to it to head away at the near post.
Hiddink's side managed to soak up the pressure and finished the half strongly when Hazard played a great ball into Bertrand Traore but the youngster's touch was poor as the ball ran through to John Ruddy.
He was called into action once again four minutes later when Cesc Fabregas' free-kick missed the head of Ivanovic but looked to be creeping into the bottom corner but Ruddy was smart enough to make the stop.
Hard done by
Neil's side felt hard done by referee Lee Mason as they felt the Blues should have been penalised for Gary Cahill's suspected back pass to Courtois towards the end of the half, but would have felt enraged when Chelsea doubled their lead in extra-time.
It was good passing play from the away side with Traore finishing off the move with a clever flick to Costa, who looked to be in a offside position, but the flag stayed down and the Spaniard did well to flick it over Ruddy - much to Neil's disgruntlement the goal stood.
Right back in it
Despite the second goal just before the break, Norwich remained positive as the second half got underway and got one back almost instantly.
Hoolahan had been linking well with Jerome all evening and looked to have laid the perfect opportunity on a plate in the 53rd minute for the striker, but he crashed it over the bar from six yards out.
They continued to battle for a way back into the match and they were finally rewarded when they got their first goal of the evening. Hoolahan was at the heart of the attack once again as he picked out Redmond, who gave Courtois no chance as he rocketed the effort into the roof of the net to make it an interesting last 20 minutes.
Unfortunately for Norwich, it wasn't to be, as they failed to create a clear cut chance to equalise in the final minutes, Martin squandering one half chance that did come along. An important win for Chelsea though, who are slowly easing up the table.