The hosts broke the deadlock just seven minutes in as Ander Herrera moved forward to the halfway line before releasing Marcus Rashford, who raced clear to place the ball calmly past the advancing Begovic with a right-footed finish.
Despite having been barracked with chants of "Judas" during United's 1-0 defeat at Chelsea in the FA Cup last month, Mourinho said victory over his former club carried no special significance.
Manchester United handed Jose Mourinho a memorable victory on his return to Chelsea as they carved out a 2-0 win at Old Trafford on Sunday.
Chelsea still lead the standings with 75 points but are now threatened by the Tottenham Hotspurs, with 71 points and in second place.
Manchester United manager Jose Mourinho has revealed that he left Zlatan Ibrahimovic out of the starting line-up for today's Premier League game at home to Chelsea in order to rest him.
The second is they [Chelsea] are the leaders and when you beat the leaders it is obviously a very good feeling.
The ruthless attacking play that has seen Chelsea score 65 goals this season was non-existent as United soaked up plenty of possession without really being troubled.
DH Ramirez leaves game with left leg injury
He'd walk another in the fourth, but quickly erased that by stabbing a comeback liner and doubling off the runner at first base. That didn't go as well, as Farquhar loaded the bases with one out on two singles (one an infield squibber) then a walk.
Chelsea's trip to Goodison Park is the most hard of their six remaining fixtures - Everton are flying at home, where they have won seven league games in a row. That half chance from Chelsea did not affect Manchester United and a minute later, at the other end, Young had another chance but fired way above the goal. Chelsea didn't have a shot on target in a league game for the first time in almost a decade. They are named Chelsea, but they could be named Arsenal or Tottenham.
The win extends fifth-placed United's unbeaten league run to 22 games and keeps them in the hunt for a Champions League spot with Mourinho seeing signs of progress from his squad.
Costa became embroiled in penalty-box tangles with United centre-backs Eric Bailly and Marcos Rojo and was booked for a late challenge on Paul Pogba.
It arrived when Jesse Lingard stole the ball from David Luiz and set up Marcus Rashford on the edge of the area, only for the striker to scuff an effort wide. That would match what Chelsea did earlier this season and would tie a record for most consecutive wins.
Chelsea manager Antonio Conte sent on Cesc Fabregas for Victor Moses, with Pedro moving to wing-back, but still the league leaders toiled, Pedro twice curling shots over De Gea's bar. At one stage four weeks ago after a 2-1 win over Stoke, Chelsea held a 13-point lead.
The Blues' run-in also looks a little kinder, with home games against Southampton, Middlesbrough, Watford and Sunderland - but a real test to come at Everton.
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