But they had to come from behind after Stephen Hunt put the Tigers ahead in the 27th minute at Stamford Bridge to cap his debut after signing from Reading on Thursday.
But 10 minutes later Chelsea levelled when Drogba's superb dipping free-kick beat the despairing dive of Boaz Myhill.
Two minutes into stoppage-time, Drogba netted the winner with a exquisite chip from a tight angle to the left of goal.
