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A stunning Che Adams-inspired Sheffield United comeback was not enough as Christian Eriksen's 88th minute goal saw Tottenham win 3-2 on aggregate in the Capital One Cup Semi-Final.
Nigel Clough's cup specialists had rescued a seemingly impossible position in the tie as substitute Adams scored twice within five minutes of coming on at Bramall Lane to level the aggregate score at 2-2.
But Eriksen, who had opened the scoring on a night of snow and hail with a brilliant free-kick, had the final say just as the Blades began to entertain hopes of another remarkable cup win.
League One United had gone into the game having beaten five Premier League teams over the previous year and they sensed another scalp despite trailing 1-0 from the first leg.
Adams, 18, gave them hope just when it seemed Spurs had it won, but the Premier League class eventually told.
Spurs had the first opening when Erik Lamela fired wildly over but it was the
Blades who had the first serious chance when visiting goalkeeper Michel Vorm fumbled a cross under pressure.
Jamie Murphy was quick to pounce but former Blade Kyle Walker got back to block on the line.
The game had a nice tempo to it and both sides looked to attack, although quality was initially lacking.
Jamie Murphy fired a cross into the Spurs area but Marc McNulty could not get near while Harry Kane shot straight at Mark Howard at the other end.
Ryan Mason also shot at Howard before Kane found the side-netting with another attempt.
Spurs claimed a seemingly 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.
Spurs immediately pushed for a second and Kane, who might have had an early hat-trick on another night, saw a low shot beaten away by Howard after a good run into the area.
But United were far from being run ragged and McNulty created an opportunity for himself as he flicked up and volleyed from 20 yards but Vorm was not tested.
Spurs looked like taking a firm grip early in the second half as Lamela and Kane caused further problems for the home defence, and there was a let-off when the latter slipped as he threatened to shoot.
Kane extended Howard again with another long-range shot and Eriksen, to a feeling of anti-climax in the away end, hit a free-kick into the wall. Mason was then denied a shooting chance by a fine Jose Baxter tackle.
But 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 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 stadium was rocking at this point and the hosts sensed a famous victory.
Their hopes almost came through six minutes from time as another substitute, Louis Reed, bust through - only to fire narrowly over from the edge of the box.
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.
Sheffield United are looking to continue their giant-killing trend in tonight's League Cup semi-final second leg. They will book themselves a place in the final if they beat Tottenham Hotspur at Bramall Lane. And manager Nigel Clough is expecting a bumper turnout.
Sheffield United manager Nigel Clough believes that his side can still book a place at Wembley despite losing their Capital One Cup first leg semi final at Spurs 1-0.
The Blades managed to hold out until the 73rd minute when Jay McEveley handled the ball under pressure from Roberto Soldado and Andros Townsend put away the resulting penalty.
Clough said: "We're still in the tie. Only that one-goal deficit, disappointed at that, I thought we were on for a nil-nil.
"We were comfortable up to half-time, just a mistake (from McEveley), it caught his arm, a bit unfortunate. He thought the ball just checked up but it's definitely handball. He didn't mean it, it's unintentional but it's certainly handball."
Clough says that they will enter the return leg with plenty of hope, adding:
"We're going to give it everything we can."
Spurs head coach Mauricio Pochettino also paid tribute to a valiant effort from the Yorkshire outfit.
"I think that Sheffield played very aggressive in the first half and cut every action and played narrow and deep too, that was difficult for us," he said.
"We didn't find the best positions to create more chances.
"I think that was a very, very difficult game but we were aware of that. Now we need look forward and try to win the second game in Sheffield."
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The Blades have already beaten Premier League sides West Ham and Southampton in the competition this season.