Chelsea's Oscar Emboaba (11) celebrates with his team-mates after scoring his side's first goal during the English Premier League soccer match at Carrow Road, Norwich, Sunday (Photo:AP)
Chelsea took a fourth-minute lead with Oscar's first-time shot from a Demba Ba lay-off.
Norwich kept a dominant Chelsea at bay for the rest of the first half before stepping up their game after the break with the reward coming in the 68th minute through Anthony Pilkington.
Just as the game looked to be heading for a draw with five minutes to go, substitute Hazard made the most of a quick Chelsea break to put the visitors back in front, while a minute later Willian curled in a superb shot on his league debut.
Chelsea manager Jose Mourinho took the credit for his substitutions after leading his side to their first away win in the league this season.
"When I make a change, the change is always fantastic and the players make it fantastic or make it rubbish - it's up to them," the BBC quoted him as saying.
"I'm happy for Willian because he is starting with us, he is not 100 percent. He was injured in the week and the kid made himself available to try to help, and when he came on he gave us a push."
Without the suspended Fernando Torres after his red card against Tottenham Hotspur last weekend, Chelsea gave only a second start of the season to striker Demba Ba.
He made his presence felt almost immediately when he got on the end of a long ball from Frank Lampard to control it well and set it up for Oscar on the edge of the area and the Brazilian fired a low shot past keeper John Ruddy for the early lead.
Ba had a chance for a goal of his own soon afterwards when he was put through by Juan Mata but Ruddy kept out the Senegal international's left-footed effort.
Chelsea could not find another goal that their first-half dominance deserved and Norwich's belief grew with chances falling to Jonny Howson and Sebastien Bassong before the break.
Norwich continued to threaten in the second half as Chelsea clung on desperately to their slender advantage but the hosts finally got the goal they had earned.
Martin Olsson's cross from the left was headed across the goal by Ricky van Wolfswinkel and Pilkington beat Chelsea defender David Luiz to head in from close range.
The equaliser seemed to shake Chelsea awake and the introduction of Hazard and Willian changed everything.
Chelsea broke quickly after a Norwich corner and Hazard's shot was pounced on by Ruddy but the ball carried on past the keeper and over the line in almost slow motion on 85 minutes.
Another counter-attack wrapped up the win a minute later as Willian curled in a first-time shot of the kind of class the club will have been hoping for from the 30 million pounds ($48.22 million) they paid for him in the close season.
"If you look at the third goal, it is a wonder strike, they've got players that can do that," Norwich manager Chris Hughton told Sky Sports.
"But the second goal, we weren't set up well as a team and their power and pace has got the better of us."
The only black cloud for Chelsea, and possibly for England ahead of their World Cup qualifiers on Oct. 11 and Oct. 15, was the sight of left back Ashley Cole being substituted in the second half with what looked like a rib injury.
In other action, Southampton continued their good start with a 2-0 win at home to Swansea City with Adam Lallana netting in the 19th minute and Jay Rodriguez adding another on 83 minutes.
Later in the day, Arsenal will have the chance to return to the top of the standings when they travel to West Bromwich Albion, while Tottenham Hotspur entertain West Ham United.
Liverpool lead with 16 points from seven matches after beating Crystal Palace 3-1 on Saturday, while Arsenal have 15.
Chelsea's win lifted them to third on 14, with Southampton level on points in fourth, one point above Manchester City and Spurs. Champions Manchester United are ninth with 10 points.
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