A pair of late goals secured three points for Jose Mourinho's side after Anthony Pilkington had cancelled out Oscar's early opener.
Jose Mourinho's side, which secured its first away win in the Premier League this season, looked to be heading for a draw after Anthony Pilkington had cancelled out Oscar's fourth-minute opener.
But Hazard's 85th-minute goal, which was aided by poor handling from John Ruddy in the 85th minute, was added to 60 seconds later by Willian, whose first-time, left-footed curled effort from the right edge of the area wrapped up the three points for Chelsea.
With Fernando Torres suspended, Demba Ba handed only his second start of the season, and the 28-year-old had an early penalty claim turned down after colliding with Sebastien Bassong inside Norwich's area.
Ba continued to threaten the hosts' defense, and he was able to assist his side's opening goal as he controlled a long ball before laying off to Oscar, who drilled his effort low past Ruddy.
Ba almost instantly doubled that lead, forcing an excellent save from Ruddy with a side-footed shot from a tight angle. Norwich carved out a presentable chance of its own in reply, with Petr Cech keeping out Jonny Howson's attempt from close range.
Norwich also had a promising chance before the end of the first half, but Howson could not engineer a shooting opportunity after Ricky van Wolfswinkel headed the ball on into a dangerous area.
Chelsea had been largely comfortable since taking the early lead, but the Blues were given a scare just a few moments after the restart following an incident that set the tone for the second period. Ramires' challenge on the breaking Pilkington had the home fans appealing for a penalty, but, yet again, referee Neil Swarbrick waved away the protests.
Van Wolfswinkel then went close to getting himself on the scoresheet, but he could only chest a bouncing ball wide with Cech out of position; however, Norwich did get the goal its endeavor deserved after 68 minutes.
Martin Olsson's cross from the left found the head of Van Wolfswinkel, and Pilkington succeeded where his Dutch teammate had previously failed by getting ahead of David Luiz and guiding the ball into the back of the net.
Both managers made attacking substitutions in a bid to claim the win late on, and it was two of Chelsea's new additions who provided the decisive contribution.
When Alexander Tettey's failed attempted clearance from Oscar's pass fell into Hazard's path, the Belgian still had plenty to do - and his shot looked destined for the safety of Ruddy's arms. The Norwich goalkeeper allowed the ball to squirm under his body and trickle into the net, though, giving Chelsea the lead in the 85th minute.
Willian then scored his first Premier League goal since his move to Chelseaa minute later as his first-time effort from the edge of the area curled over Ruddy, who could not get close to the strike.
The win fires Chelsea to third in the table, two points behind league-leader Liverpool, while Norwich remains on seven points, two above the relegation zone.