The Belgian fired the hosts in front shortly before the break in a half largely dominated by Louis van Gaal’s side, who nearly took an early lead through Wayne Rooney.
It could have been 2-0 after the break as Hazard hit the crossbar from close range, while Rooney saw another effort go wide and Radamel Falcao hit the post as the visitors saw far more of the ball. Ander Herrera thought he had won a penalty in stoppage time only to be booked for diving.
And a resilient Chelsea dug in to take victory, the result moving them 10 points clear of Arsenal and 11 above United, putting the Blues within touching distance of their first league title in five seasons. They can become champions with wins in their next two games.
It was a confident start from United, who were clearly looking to take control of possession and play through Chelsea.
They should have been in front inside four minutes as Luke Shaw raced forward before pulling the ball back to the edge of the box for Rooney, who had plenty of time and space but could only pull his shot wide of the target.
The effort initially looked like it had gone in, and it fooled both the travelling fans and David de Gea at the opposite end, prompting much derision from the Chelsea supporters.
De Gea almost suffered further embarrassment as a ball came in from the right and the goalkeeper raced to get to it in front of Hazard. The ball appeared to hit the goalkeeper’s arm outside the box, but it wasn’t spotted by the officials and the Spaniard escaped punishment.
Despite United’s dominance on the ball, chances were at a premium, and it wasn’t until the 38th minute that the first shot on target was registered as Chelsea took the lead.
The ball was played in from the left for Oscar, who produced a delightful backheel through the middle for Hazard, and the Belgian charged into the area before firing a left-footed shot underneath De Gea.
The visitors were angry with the strike, however, as John Terry appeared to foul Radamel Falcao in midfield in the build-up.
But the goal stood as the pattern of play continued in the same vein after the break, although it was Chelsea who nearly doubled their lead within the first 10 minutes of the half. Didier Drogba was played through and his shot looped over De Gea thanks to a deflection off Chris Smalling, but Hazard could only find the crossbar with his jumping volley.
It wasn’t until the 61st minute when Van Gaal’s men finally brought a save out of Thibaut Courtois, who had to be alert to parry away Paddy McNair’s drilled 25-yard effort. Rooney then fired just wide of the target after meeting Fellaini’s knock-down, only to see it deflect off target as the visitors continued to see all of the ball.
Falcao worked hard up front for United but struggled for opportunities, his best effort of the match coming from Shaw’s pass as he then sliced a shot onto the post from a tight angle.
Chelsea looked confident of victory as the game moved into stoppage time, but referee Mike Dean had to make a key decision as Herrera charged down a loose ball in the box before tripping over Gary Cahill’s leg.
The referee blew the whistle straight away but booked the United midfielder for simulation, and the hosts held on for a key victory to move themselves even closer to the Premier League title.
Chelsea: Courtois (6), Ivanovic (5), Terry (7), Cahill (7), Azpilicueta (6), Zouma (8), Matic (7), Oscar (6), Fabregas (6), Hazard (8), Drogba (5).
Used subs: Ramires (6), Mikel (6), Willian (6).
Manchester United: De Gea (5), Valencia (6), Smalling (6), McNair (7), Shaw (8), Mata (7), Rooney (5), Herrera (6), Young (6), Fellaini (6), Falcao (6).
Used subs: Blackett (5), Di Maria (6), Januzaj (6).
Man of the match: Eden Hazard