Harry Kane said it "means the world" for Tottenham to reach the final of the Capital One Cup.
Christian Eriksen's double ensured Spurs avoided extra time at League One Sheffield United, his late goal giving the Londoners a 3-2 win on aggregate.
They will meet Chelsea at Wembley on 1 March and Kane told Sky Sports 1: "It means the world.
"We were in cruise control and that was the worst thing we could have done. Credit to Sheffield United but it shows how much character we have."
Che Adams scored two goals in 140 seconds for United to put them 2-1 ahead on the night and level at 2-2 overall after Eriksen's opener.
Extra-time looked likely at that stage but Eriksen made sure Spurs will be facing Chelsea in a month.
"At Wembley anything can happen," he added. "It's 1-1 between us this season, it's a decider if you like and there's no better place for it than Wembley."
Eriksen himself said: "When they scored they got confidence, and you saw the crowd, but we had to play our best.
"To be in the final is what everyone wants at Tottenham."
Losing manager Nigel Clough could not fault his players.
"I'm incredibly proud of the players, to run a team like Spurs so close, that bit of quality made the difference," he said.
"We knew one goal would change the complexion. We got the second and then had a chance for a third and just missed it; I'm very proud."
Mauricio Pochettino was happy to see his Spurs players weather the storm.
"Happy for our players, they deserve to go to the final, Wembley, and happy too for our supporters. They deserve it too, I'm very happy," he said.
"In this type of game it's important to kill it and score more than one; this was a difficult game and there was unbelievable pressure.
"The last 12, 10 minutes were difficult but I think we deserved it."
PA
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