"They showed that they have improved in their games against Manchester United and Arsenal - and Chelsea are another of these teams.
"They have shown they are able to concentrate and fight with the best of them."
Woodgate, who signed for the north London side from Middlesbrough last month and was playing only his fourth game for the club, popped up with the headed winner four minutes into the first half of extra-time.
"It's a great feeling to win our first trophy as a team - we've got a great team spirit and work ethic," said the 28-year-old defender.
"Scoring that goal gave me one of the best feelings ever."
I hope this is the start of something big
Spurs coach Gus Poyet
Goalkeeper Paul Robinson, who pulled off a fantastic save with his feet to deny Salomon Kalou in extra-time, said he was delighted to have picked up a winners' medal for the first time in his career.
"It means everything," said the Spurs shot-stopper.
"I've not won anything in the game so to land this trophy after beating such tough opponents is great."
He added: "The Kalou shot was a bit of a blur - I just wanted to keep it out."
Spurs first-team coach Gus Poyet said he hoped the new-found belief in the side would lead them to greater success.
"We needed to be better than Chelsea and I think we played better and believed a little bit more," said the former Blues and Tottenham midfielder.
"This team is unbelievable. I hope this is the start of something big."
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