The start of the match was marred by a small section of Chelsea supporters, who disrupted the minute's silence held in honour of the victims of Hillsborough as well as Italian footballer Piermario Morisini.
When the action got underway, Chelsea had the best opening chances as Frank Lampard looped his weak header over the bar.
Drogba then had two chances in quick succession, but the striker flashed his free kick over before scuffing his volley.
Scott Parker's effort was blocked by the onrushing Lampard in the penalty as Spurs soon began to take control of the game.
However, Chelsea continued to look a threat in attack as Juan Mata nearly got on the end of Salomon Kalou's pass.
Harry Redknapp's side nearly broke the deadlock five minutes before the break, but Jonh Terry managed to clear Rafael van der Vaart's header off the line after good work from Emmanuel Adebayor.
Van der Vaart came close almost instantly when his cross missed everyone to hit the crossbar.
The only goal of the half came with two minutes to play as Drogba shook off the attention of William Gallas to lash the ball in past Carlo Cudicini.
Click here for Sports Mole's live commentary of the match from Wembley.