The London side dominated the first half, rewarded with a stunning opener from Joe Cole, and despite United improving after the break, Drogba came off the bench to score what turned out to be the winner.
The Ivory Coast forward was played through by Salomon Kalou and as he crashed the ball into the net, the United defence stood bewildered after Drogba appeared to be in an offside position.
"You expect officials to get that one right," United's assistant manager Mike Phelan added. "It was not even close. Their job is to keep their composure and they failed to do that."
United also had a penalty appeal turned down by referee Mike Dean when Ji-Sung Park was challenged by Yuri Zhirkov, while the visitors were denied a spot-kick when Gary Neville appeared to barge into Nicolas Anelka.
Officials got the big decisions wrong - Phelan
Dean was in the headlines this week after Blackburn's Martin Olsson admitted he deliberately went down in the box at Burnley because of the referee's reputation for giving penalties.
Asked whether Dean's decisions had been affected by this, Ferguson said: "I don't know about that but when I saw Mike Dean I did worry, I must admit."
Meanwhile, Chelsea boss Carlo Ancelotti remained unconcerned with the controversial decisions.
"I never comment on the referee's decision," the Italian said. "It is offside when the linesman's flag goes up, it is a penalty when the referee blows his whistle."
Chelsea's goalscorer Cole, enjoying a run of good form and a place in the starting line-up, added: "I felt the referee did a good job. With the pace of the game being so quick these days everybody's going to make small mistakes."
With the loss coming on the back of United's lacklustre defeat in the Champions League to Bayern Munich in midweek, Ferguson admitted his players lacked an edge in the opening 45 minutes.
The Scot added: "The first half we looked leggy, it took us too long to get going and Chelsea were by far the better team.
Battling victory delights Ancelotti
"The second half we did well, we were unfortunate not to get something from the game. We dominated it in the second period but we just didn't get there."
Ancelotti, whose decision to start Drogba on the bench was vindicated with the second goal, was delighted to pick up three invaluable points in the race for the title.
"We kept a high intensity in the first half. We had good control of the game," Ancelotti said.
"We trained very well during the week because we didn't have a Champions League match. You lose a lot of energy.
"We played very well in the first half. It's a very important victory for us and a very important performance. To win here is not easy, we did a good job."
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