Drogba came as a second-half substitute and scored his team's equaliser but he fluffed the chance to secure a dramatic victory when his weak spot-kick was saved by Heurelho Gomes.
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The Spurs goalkeeper had been at fault for Chelsea's goal, allowing Drogba's shot to squirm through his grasp, and he gave away the penalty with a clumsy foul on Ramires.
"We deserved to win the game, especially in the second half, we dominated them in the second half and that's more like the Chelsea we all know," Chelsea captain John Terry told Sky Sports.
"Didier was brave enough to stand up and take it. You miss some you score some it was just one of those things. Playing like we did in the second half everything looked back to normal.
"The hunger and spirit and determination was much better today and that's what we want from a Chelsea side."
"They pressed us in the second half, we were 1-0 up and we tended to take a backward step. I said we had to keep being positive and not step back because I thought we could score again in the second half," Spurs manager Harry Redknapp said.
"But credit to them they don't give up, they're champions and they kept working hard and in the end it was a fair result."
However, the Russian's first touch took him clear of two defenders and he planted a low shot beyond Petr Cech.
Drogba beat him to a long Cech clearance and was quicker to turn on to the loose ball but Gomes should still have saved the Ivorian's shot straight at him.
