Woodgate said: "I don't really go up for corners but I took a chance and was able to get my head on it and luckily it went in.
"I think we were the best team and hopefully we can push on from this next year. Chelsea are a top side and they showed that today but we did it."
Spurs manager Juande Ramos added: "I am happy because the people (fans) are very happy."
Ledley King hailed the "spirit" of his side after they battled back from a goal down to claim the trophy.
"Its amazing. It's been a long time to get here - too long. The only other time we got to a final we lost and we didn't want that to happen today," said King.
"You saw the commitment levels from the players today and we showed we're a good team when we got going.
"It is tough being 1-0 down. Their goal came from a free-kick but I thought they didn't really create much more than that.
"The spirit was fantastic and at 1-0 down a lot of teams would have folded but we were fantastic to come back and win.
"We have some good players here and hopefully we can use this win and move forward."
Avram Grant lamented the manner in which Chelsea lost, but maintained there was still plenty to play for this season.
Grant, who replaced Jose Mourinho as manager last September, said: "We did not start the game very well, but then after 25 minutes or so we progressed and scored a goal.
"Then in the second half, Tottenham did not create many chances until the penalty decision, which brought them back into the game.
"The ball may have hit Wayne Bridge in the hand, but it was still a rash decision - like in the last seconds of the match to stop the game when [Salomon] Kalou was right in front of the goal.
"For the penalty, it was not even the referee's decision. It was the linesman.
"You can think it over and know that every time the ball touches a player's hand it is not a penalty. If it is deliberate yes, but if not then no."
Grant added: "The players are upset because the referee stopped the game when John Terry had headed the ball, and Kalou was in front of the goal. Then he stopped it.
"How can he decide it was not one or two seconds more? I do not understand that.
"However, we cannot take the time back. We have lost the final and that is it. We have to learn from it."
Grant refused to blame Petr Cech's error - his miscued punch bounced back off Woodgate to give Spurs the victory - for the defeat.
He said: "We win as a team and lose as a team.
"There are always personal mistakes in football, but we lost as a team.
"We will see what we can learn from this game."
Grant - whose side are still in both the Champions League and FA Cup, while not yet out of the race for the Barclays Premier League - added: "We drew at 90 minutes again today. We would like to win and we'll think about everything.
"We would like to get back to winning a few games in a way.
"We got to the final, which was not easy.
"Now, we have lost but we are still in all three other competitions and will do our best in them."