Even after Michael Ballack's late penalty winner, United still had two chances cleared off the line by a desperate Chelsea rearguard in an amazing second-half at Stamford Bridge.
It was Ballack who nodded Chelsea in front just before halftime.
The Germany captain was left unmarked at the far post to head home Drogba's cross in the 45th minute.
Joe Cole had rapped the United crossbar earlier in the half while United, who lost defender Nemanja Vidic with a facial injury, failed to test Petr Cech at the opposite end.
United got themselves back into the game after the break thanks to a monumental blunder by Carvalho.
The Portuguese defender passed the ball straight to Wayne Rooney, who accepted the gift by firing beyond Petr Cech into the net.
Drogba then clashed with Ballack over the taking of a free-kick. Ballack stormed off, leaving Drogba to take it and Edwin van der Sar pulled off a magnificent save. The row continued during treatment to John Obi Mikel, with Drogba firing verbals at both Ballack and Blues coach Steve Clarke.
Then in the final five minutes, Chelsea were awarded a penalty after Michael Carrick was penalised for handling in the area. Ballack stepped up and converted coolly - and among the first to congratulate him was Drogba.
But the action was still not over as first substitute Cristiano Ronaldo saw his left-foot drive cleared off the line by Ashley Cole before Blues striker Andriy Shevchenko did the same to block Darren Fletcher's injury-time header.
Chelsea's victory brings them level on 81 points with leaders United, who remain top on goal difference with now two games to play.