Wilder referenced the attitude, application and desire of Liverpool after the Blades' 2-0 loss to the Premier League leaders at Anfield and Woodgate feels his own side showed those qualities in a richly-deserved draw with Jose Mourinho's Spurs, which has earned them a replay and first-ever trip to the Tottenham Hotspur Stadium next week.
Woodgate, a former Spurs player, said: "It was just the team performance, the energy of the team and the battling and fighting.
"I saw Chris Wilder’s interview the other day against Liverpool where he spoke about doing the basics of the game, running, fighting, second balls, winning your headers and winning your tackles.
"They are the fundamentals of the game and that was the biggest thing I was proud of, the team fighting for the shirt."
On earning a replay, he added: "It is going to be a really difficult game. But I will not be thinking of that, I will be thinking how we can win the game.
"Tottenham are a really incredible team, but you never know. In this life upsets do happen, it has been proven over the years, so let's hope we upset the apple cart."
Meanwhile, Woodgate has confirmed that summer signings Marcus Browne and Marc Bola will head on loan this month to enable them to get some more game time after struggling for opportunities of late.
Woodgate continued: "I know we have had a lot of enquiries for Marcus Browne and Bola, so let’s see what the best place is for them to play games.
"We do it when we are looking at players, so we need that now with Marcus Browne and Marc Bola."