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 March 1 and Kane said afterwards: "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's time.
"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."
Two-goal hero Eriksen said: "Sheffield United definitely gave us a big fight. They gained confidence after they scored their first goal, the second one was a bit lucky but we came back at the end.
"This season when we have been in front we have given stupid goals away and it is something we need to stop doing. Chelsea will probably want revenge in the final but I am really looking forward to it."
United boss Nigel Clough said: "I am incredibly proud of the players and what they gave over the games. It was just a bit of quality that made the difference. Eriksen's two goals tonight were superb.
"When we were a goal down we knew one goal would change the complexion of the game and the crowd really got into it. We are delighted with Che Adams tonight, we like the look of him. But he is still raw and has potential.
"We will take lots of positives and when we get the crowd behind us we are a force at home."
Spurs claimed a decisive edge in the tie just before the half-hour after Benjamin Stambouli was brought down on the right.
There was little doubt Eriksen would go for goal from the set-piece and he did not disappoint, curling a delicate shot over the wall and in off the far post.
Just as it seemed the tie was drifting away from the Blades, Clough turned it on its head by sending on Adams for Jamal Campbell-Ryce after 74 minutes.
Within three minutes he had levelled the score on the night as he appeared in the box at the right moment to turn in a low cross from Ryan Flynn.
Two minutes later the teenager was at it again. This time Murphy got to the byline and pulled the ball back for Adams to strike a shot that took a deflection off Eric Dier and flew in.
The possibility of extra-time began to loom but Eriksen came to the fore again to rescue the London side.
Kane picked him out with a superb pass and the Dane showed great footwork and composure to finish with a low shot and send Spurs to a Wembley date with Chelsea on March 1.