Joe Cole scored Chelsea’s first after 20 minutes before substitute Drogba made it 2-0 after 79 despite being offside when he collected the ball.
Substitute Federico Macheda then handled the ball when he pulled one back for United two minutes later.
Arsenal stayed in the title race by beating Wolverhampton Wanderers 1-0 at the Emirates Stadium thanks to a header four minutes into stoppage time from substitute Nicklas Bendtner.
With five matches to play Chelsea have 74 points, United 72 and Arsenal 71.
Manchester City rose to fourth, which brings a spot in the Champions League playoffs, by crushing Burnley 6-1 at Turf Moor. Tottenham Hotspur slipped to fifth after losing 3-1 at Sunderland.
City made the best start to a Premier League match by any team for 17 years with three goals in the opening seven minutes from Emmanuel Adebayor, Craig Bellamy and Carlos Tevez.
Patrick Vieira, Adebayor and Vincent Kompany then completed the rout on a quagmire of a pitch.
While it was a good day for City, it was a bad one for the red half of Manchester.
“Chelsea are in the driving seat now, there’s no question about that,” United manager Alex Ferguson told his club’s website (www.manutd.com) before complaining about Drogba’s goal.
“What I don’t understand is the linesman’s directly in front of it. He has no one near him and he gets it wrong. A game of that magnitude, you really need quality officials and we didn’t get them today.”
Chelsea, inspired by an outstanding display from Florent Malouda, had the better of the opening half. United, without injured striker Wayne Rooney, improved in the second in a tense game with few scoring chances for either side.
TITLE FIGHT
The match was the first of two titanic encounters in Manchester yesterday, with the WBA heavyweight title fight between champion David Haye and John Ruiz taking place in the city later in the evening.
Haye was presented to the crowd ahead of kickoff before Chelsea started the game with real verve.
United looked leaden-footed and flat following their defeat by Bayern Munich in the Champions League on Tuesday and Chelsea’s early dominance was rewarded with a goal.
Malouda, who has shone recently, powered his way through three defenders before holding off Darren Fletcher and crossing for Cole to score with an impudent flick off his heel.
United looked livelier after the break with Dimitar Berbatov leading the line and going close with a header after 60 minutes.
Just when United seemed likely to find an equaliser Chelsea doubled their lead.
Drogba, who had replaced Nicolas Anelka 10 minutes earlier, blasted the ball past `keeper Edwin van der Sar but he was clearly in an offside position when he collected the ball from Salomon Kalou.
Italian teenager Macheda then set up a frantic finale when he bundled the ball in from close range.
Arsenal scored their third injury-time winner in six matches to beat Wolves.
The Gunners were made to fight by Wolves who had Karl Henry sent off after 66 minutes for a foul on Tomas Rosicky, a red card Arsenal manager Arsene Wenger said was harsh.
Arsenal’s north London rivals Tottenham slipped up at Sunderland where Darren Bent scored in the first minute before making it 2-0 with a penalty in the 29th.
Bent, who also missed a spot kick when Sunderland played at Spurs in November, then missed two more penalties before Anton Ferdinand had a goal disallowed for Sunderland.
Peter Crouch pulled a goal back for the visitors with 18 minutes to go before Boudewijn Zenden volleyed in Sunderland’s third.
“It was one of the all-time great volleys,” said Spurs manager Harry Redknapp.
Bottom club Portsmouth drew 0-0 with Blackburn Rovers while Stoke City defeated relegation-threatened Hull City 2-0.
