It seemed that a draw was on the cards until Chelsea's replacements both struck within the last five minutes.
Below, Sports Mole has looked back over the 90 minutes to determine if the result was a fair one.
On target 3
On target 8
Was the result fair?
Had Norwich claimed a draw from this one, Chelsea could have had few complaints. The Premier League has habit of punishing mistakes, though, and Norwich found that out today. A share of the spoils may have been fair, but if either side deserved to win, it was Chelsea.
It took the Canaries a while to settle and at times during the first half they rode their luck. However, there was a positive transformation in the second half and they got themselves among the Chelsea players, which forced mistakes. When they levelled proceedings, there was a point when Chris Hughton's side looked the more likely to score the winning goal. As it was, a mistake cost them dearly, but once the disappointment subsides, there were reasons to be positive for Hughton.
Not that he needs much of an ego boost, but Jose Mourinho proved his worth today. Despite some scepticism, it was the right decision to start with Demba Ba, who set up the opening goal. Much later, the winning goals were scored by two of Mourinho's substitutes. In terms of the balance of play, Chelsea controlled virtually the entire first half, but Mourinho may well be worried by how his side let Norwich back into proceedings after the restart. Still, they rallied late and got the victory.
Sports Mole's man of the match
Oscar: The technique that he showed to score the opening goal was reason enough to hand the Brazilian this award. The goal aside, he was lively and always looked to exploit any pockets of space between the Norwich defence and midfield. With that sort of display, it is not hard to see why he appears to be a favourite of his manager.
It will probably take Alexander Tettey some time to get to sleep tonight. He appeared to have the situation under control as Chelsea broke with the score at 1-1, but his touch was far too heavy and it presented a chance to Hazard. The Belgian took full advantage by drilling the ball underneath John Ruddy, putting his side 2-1 up in the process with just five minutes left to play.
A strange outing by Neil Swarbrick, who during the first 20 minutes appeared to have lost his whistle. Numerous fouls were committed, but he decided against awarding free kicks. In the second half he took an opposite approach, much to frustration of both sets of players. Add to that the fact that he failed to award Norwich a penalty when Pilkington had been upended by Ramires and it's fair to say it wasn't the official's finest display.
Norwich: Next time out Norwich will travel to the capital to face Arsenal.
Chelsea: The Blues return to home comforts after the international break when they play host to Cardiff City at Stamford Bridge.