Former Chelsea striker Romelu Lukaku scored and then created the winning goal against his old team at Old Trafford on Sunday.
The win put United back in second place in the Premier League, still 13 points behind neighbors Manchester City, while Chelsea drop below Tottenham into fifth place.
The win was just about merited after a much improved second- half performance by Jose Mourinho's side.
Chelsea were the brighter team early on and Alvaro Morata, brought back into the side, volleyed against the bar after only four minutes.
The Londoners deserved the lead that Willian gave them after 31 minutes, taking Eden Hazard's pass and beating David de Gea at his near post.
As United improved, Lukaku belied his reputation for not scoring against the big clubs by equalizing before halftime, then crossed perfectly for substitute Jesse Lingard to head the winner in the 75th minute.
Each manager had brought an expensive signing back into their starting XI with Morata replacing Pedro for Chelsea and United's record signing Paul Pogba picked, although youngster Scott McTominay kept his place.
Morata was more influential initially, hitting the crossbar from Marcos Alonso's volleyed cross.
Alexis Sanchez failed to convert a good chance for United, who fell behind four minutes later.
Willian, the scorer against Barcelona in midweek, led a break from his own half and took Hazard's pass before a shot that De Gea should have done better with.
In the second half the goalkeeper was also troubled by drives from Danny Drinkwater and Willian that bounced in front of him.
Chelsea's Thibaut Courtois was more impressive in saving a spectacular effort from Lukaku from a pass by Sanchez.
United were getting on top and Lingard had been on for only a few minutes when he headed in a perfect curling cross from the right by Lukaku, whose performance added to Chelsea's frustration at not having secured his return from Everton in the close-season.
Their mood was not improved when Hazard had to come off and Morata's late effort was ruled out for offside by the narrowest of margins.