Drogba put the blues ahead with a great free-kick just six minutes before half-time but then Dimitar Berbatov equalized from the penalty spot on 68 minutes and Tottenham's new signing Jonathan Woodgate headed the winner late on, four minutes into the first period of extra time.
This was spurs first trophy since 1999.
Tottenham striker Robbie Keane was lost for words after the game.
"I don't normally go for corners, but I just took a chance and put my head on it," an ecstatic Woodgate said.
"I think we were the best team. They were a tough side, but we have done well to come out on top."
"It's massive, I'm speechless," he said. "This is a dream come true for me. I have dreamt about lifting that Cup. It's unbelievable."
Chelsea boss Grant said after the game... "I think even in the second half Tottenham were not in the game and that we were the better side," Grant told Sky Sports 1.
"The penalty put them back in the game and that turned the match. The second goal was from a set piece so we are disappointed.
"I did not think it was a penalty. Wayne Bridge could not keep his hand behind his back when he was challenging for the ball and you saw it was not the referee who gave it but the linesman.
"It was always going to be close because cup finals are like this and it was not a great performance from both teams. We tried to score a goal and keep it tight but we lost.''
