Chelsea are now four wins away from retaining their title, while City's season is effectively over.
Drogba's first was a fine left footer from close range while his second was a controversial finish after he clearly controlled the ball with his hand.
The Ivory Coast international actually had the ball in the net within five minutes but, after tapping the ball in from a Paulo Ferreira cross, he was pulled up for offside.
Two minutes later, a beleaguered City were lucky to escape when Eidur Gudjohnsen, in for Michael Essien, scuffed a daisy-cutter of a left footer wide of David James' left-hand upright.
James then produced a fine save from a Ricardo Carvalho pile-driver on 12 minutes as the Portuguese defender saw his 25-yard drive tipped over the bar.
The City keeper found his goal exposed two minutes later when an excellent defence-splitting pass by left-back Asier del Horno allowed Drogba to try an audacious lob from the edge of the City area. Although he saw his effort go over James' head it also drifted over the crossbar.
The one-way traffic continued on 22 minutes and, again, James was the hero - saving brilliantly with a punch away from Drogba's powerful on-target left-foot strike from just inside the penalty area.
City eventually managed to get outside their own half on 26 minutes and it produced their first effort on goal. Danny Mills, booked a minute earlier for scything down Damien Duff, raced up field and let fly from 20 yards.
His left-footer was spilled by Chelsea stopper Petr Cech but was cleared to safety.
Drogba's goal on 30 minutes finally broke City's dogged rearguard resistance. Latching onto a fine pass by Gudjohnsen, Drogba turned David Sommeil on the left of the area, before ramming the ball past a helpless James.
But there was controversy three minutes later when Drogba was guilty of blatant handball as he drilled home the second from six yards out.
John Terry's marvellous downward header was expertly cleared by James but the ball fell to Joe Cole who volleyed it back into the area.
Drogba clearly controlled the ball with his right arm before smashing his shot past James.
Distin, already booked for dissent when Chelsea scored their second goal, was then red-carded at half-time for his failure to return the ball to the ref.
City boss Stuart Pearce replaced Sommeil with Albert Riera during the break while Chelsea's first substitution came on 53 minutes - Essien replacing Duff.
Mills then forced a fine save from Cech on 54 minutes when his curling 25-yard free-kick was tipped around the goalkeeper's left upright.
A succession of chances went begging for the Premiership champions - Cole, Essien and Drogba - all guilty of poor finishing.
City for their part, were content just to restrict Chelsea to holding possession of the ball in midfield.
Shaun Wright-Phillips entered the fray to face his former club at the expense of Cole with 18 minutes remaining.
Within seconds the Chelsea wing man skipped past a couple of defenders before seeing his shot flash across the area.
Drogba missed a chance to score his third when James charged down his goalbound shot on 77 minutes - Hernan Crespo missing an open goal from the resultant loose ball, by nodding a diving header wide.
Drogba then suffered a poke in the eye, five minutes from time and failed to return to the pitch but Chelsea ran out deserved winners.