The home side went behind to a Stephen Hunt goal on 28 minutes before the Ivory Coast striker netted the equaliser on 37 minutes and then lobbed goalkeeper Boaz Myhill in stoppage time to secure all three points in the league season opener.
At times Drogba was unplayable and missed a sitter within 65 seconds when he volleyed over a Michael Essien cross.
Hull were desperate in defence but finally got in to Chelsea's last third on 14 minutes when Hunt's whipped left-wing cross zipped off the head of Dean Marney's head from eight yards out and just went wide of the post.
George Boateng forced a fine diving save from Petr Cech three minutes later when the midfielder's volley from just outside the penalty area was pushed away for a corner.
On 28 minutes Hull took the lead. After Jose Bosingwa was penalised for pulling Hunt's shirt, the resultant free-kick saw the ball fall to Boateng, whose left-foot shot deflected off John Obi Mikel into the path of Hunt and he coolly slotted home from eight yards out.
Drogba tried to remedy the situation six minutes later but his fierce rasping 20-yarder was held by goalkeeper Myhill.
But seconds later he did succeed with a superb free-kick from 20 yards out when he placed the ball with venom into Myhill's left-hand corner of the net.
Essien and Drogba both forced fine saves from Myhill with long-range efforts seconds before the break.
Ancelotti had seen enough of Mikel and replaced him with Michael Ballack at half-time.
A mad scramble on 57 minutes saw Hull survive several goal attempts, the best of which saw Myhill produce a point-blank save from Drogba six yards out.
Hull should have scored a breakaway goal but Caleb Folan made too much of a fuss of the ball and it was cleared.
It was Essien who had the next offering with a half-volley from 25 yards which whistled inches wide of the Hull goal.
Hull could have snatched a stunning victory in the dying seconds when Folan's cross was deflected towards goal and Cech was forced to tip the ball over the bar.
Michael Turner made a last-ditch saving tackle from Drogba, but seconds later the hitman's exquisite chip sailed over Myhill and into the far corner of his net to break plucky Hull's resistance.