Woodgate's fortuitous 94th minute winner separated two evenly-matched sides in an entertaining final after the scores were locked at 1-1 after 90 minutes.
The England defender was playing in only his fourth game for his new side following his move from Middlesbrough in the January transfer window.
However, after failing to reach Jermaine Jenas' initial free-kick with a header, he knew little about the winner as Chelsea keeper Petr Cech pushed it back against him and the ball ricocheted into the empty net.
Tottenham had fought back from a goal down to send the final into extra-time despite being on top for large sections of the match.
The north London club roared out of the starting block and should have been ahead inside the first minute when Robbie Keane picked up a loose pass from Juliano Belletti only to see his early shot diverted around the post by John Terry.
Chelsea coach Avram Grant had included his fit-again skipper and also Frank Lampard, but it was Spurs who continued to hold the upper hand.
Twice in the ninth minute they went close when Pascal Chimbonda headed a corner onto the top of the crossbar, before Dimitar Berbatov failed to hit the target with a glancing header from eight yards after a pinpoint Robbie Keane cross.
Spurs looked like they might rue those early misses as Chelsea gradually gained a foothold in the game and Didier Drogba fired them into a 39th minute lead.
The strangely subdued striker had gone close with an earlier free-kick from 25 yards before sending a dipping second effort over the wall and past a flat-footed Paul Robinson to break the deadlock.
Spurs seemed stunned after bossing the match up until that point and it took them until midway through the second half to recapture their earlier attacking initiative.
However, they were indebted to Wayne Bridge for giving away a needless 69th minute penalty to get themselves back on level terms when the assistant-referee correctly spotted the left-back had handled the ball under pressure from Tom Huddlestone.
Berbatov was the coolest man inside the stadium as he sent Cech the wrong way from the spot with a calm kick to the keeper's right to restore parity and set up a grandstand finish.
Didier Zokora wasted a gilt-edged chance to end the tie before extra-time when he slammed a low shot straight at Cech when through one-on-one, while the keeper also beat away a crisp Berbatov drive eight minutes from full-time.
However, he could do little about Woodgate's winner in the first period of extra-time and, although Chelsea piled forward looking for an equaliser to take the final into penalties, they failed to find the necessary quality in the final third to deny Spurs a first piece of silverware under the impressive Ramos.