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.
GEAR:Up to 16% off Chelsea kits with Kitbag. Free delivery on orders over £50
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.
Chelsea, on 31 points, have taken only six from their last seven matches but are still only one behind leaders Arsenal and Manchester City and level with Manchester United, who host Arsenal on Monday in one of their two games in hand on the champions.
"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."
Spurs are fifth on 27 with Bolton Wanderers up to sixth on 26 after they beat Blackburn Rovers 2-1 earlier on Sunday.
"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."
Spurs went ahead after 15 minutes when Jermain Defoe picked out Roman Pavlyuchenko on the edge of the box with little obvious danger.
FIRST TOUCH
However, the Russian's first touch took him clear of two defenders and he planted a low shot beyond Petr Cech.
Chelsea had more of the ball but their build-up was ponderous.
Manager Carlo Ancelotti cut a frustrated figure on the touchline, shaking his head, half-heartedly waving his players forward and getting involved in an argument with Terry as he urged his captain to hold a higher line.
Gomes was on his mettle early in the second, though, turning away a shot by Drogba and acrobatically tipping over after the ball skidded off the head of Wilson Palacios.
Spurs centre-back Michael Dawson had looked solid on his first appearance in three months but he was at fault for the equaliser.
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.
Chelsea pressed hard for the winner, b
