It took the Blues just over three minutes to take the lead in Sunday's Premier League encounter as Demba Ba, replacing the suspended Fernando Torres, held off Sebastien Bassong before teeing up Oscar to fire into the bottom corner with the outside of his boot.
After a bout of early second-half pressure, Norwich levelled through Anthony Pilkington when he headed past Petr Cech after Ricky van Wolfswinkel's knock-down had fallen into his path.
Substitute Hazard restored Chelsea's lead on the counter-attack when his effort crept underneath John Ruddy before summer signing Willian secured the points with a stunning strike.
With Torres unavailable, Jose Mourinho left Samuel Eto'o on the bench in favour of Ba, with the Senegal striker making only his second Premier League start of the season.
The Chelsea boss was looking to secure his first league win on the road since returning to the Stamford Bridge club this summer and his side got off to the perfect start thanks to the in-form Oscar.
Frank Lampard's long ball was taken down by from the air by Ba, who showed his strength, turning away from the challenge before laying off for the Brazilian midfielder to net his fourth Premier League goal of the season.
Ba was causing Norwich all sorts of problems and he almost doubled Chelsea's goal tally when he was found unmarked in the area, only for John Ruddy to palm away his strike across goal with a smart save.
The 28-year old looked like a man with a point to prove in the early stages and Bassong denied him the chance to get on the score sheet when he stabbed the ball towards an empty net having forced Ruddy to fumble John Terry's long ball.
It took Norwich until the midway point of the first half to have an effort on target as Jonny Howson almost forced David Luiz to put through his own net following van Wolfswinkel's cross, but Cech managed to divert the ball over the bar and behind for a corner.
As the half drew to a close, Luiz, who was just outside of the centre circle, saw his speculative effort land on the roof of the net after the Brazilian had spotted Ruddy off his line.
The hosts struggled to create too many real chances in the opening 45 minutes, although the last action of the half came in the Chelsea box as Bassong saw his acrobatic volley caught by Cech.
Immediately after the break, Ba came close to doubling Chelsea's advantage when he flicked Ramires' cross wide after an intelligent run across goal.
Carrow Road was then up in arms as Pilkington saw his penalty appeal waved away by referee Neil Swarbrick following what appeared to be a clip to his heels from Ramires.
Chris Hughton's men showed much more courage in the early stages of the second half as van Wolfswinkel and Pilkington both squandered half chances before Alexander Tettey blazed over from distance.
The Canaries' persistence eventually paid off after 67 minutes when Pilkington got the better of Luiz before guiding home from close range after van Wolfswinkel had nodded Martin Olsson's cross into the six-yard box.
In an attempt to reclaim the lead, Mourinho introduced Eto'o, Hazard and Willian across a 10-minute period.
The changes did pay dividends for Chelsea as Hazard took advantage of a defensive lapse from Alex Tettey before firing underneath Ruddy to put the visitors back in the ascendancy with just over five minutes remaining.
Chelsea had it all wrapped up moments later when another of Mourinho's substitutes, Willian, struck to make it 3-1 with a superb curling effort into the top corner marking his first goal since his summer move from Anzhi Makhachkala.
The win took Chelsea into third place, two points behind leaders Arsenal, while Norwich slip into the relegation zone.