The Blues went behind after 15 minutes when Roman Pavlyuchenko put Spurs in front with a precise finish. With Chelsea's attack misfiring, Drogba was called from the bench at the break and levelled with 20 minutes of the match remaining.
However, the Ivorian missed the chance to complete the job late on from 12 yards. Heurelho Gomes fouled Ramires in the box, but redeemed himself by palming away Drogba's spot-kick.
The draw means Chelsea remain in fourth spot, and although they are now level on points with United, Sir Alex's men are ahead on goals scored and have two games in hand. Arsenal, the Reds' opponents on Monday night of course, are still in first spot, while Manchester City moved up to second on Saturday with their 3-1 victory over West Ham.