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.
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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.
Giant-killers Bradford City could take on Premier League Sunderland at Valley Parade in the fifth round of the FA Cup.
They have been drawn to either Sunderland or Championship Fulham in the next round.
The Bantams pulled off one of the biggest shocks in cup history after they beat Premier League table-topping Chelsea 4-2 at Stamford Bridge.
Meanwhile, Sheffield United could face Manchester United at Bramall Lane if the Blades can get past Preston if the Reds can beat League Two Cambridge in their respective replays.
Bradford City manager Phil Parkinson hopes his side's victory over Chelsea will be rewarded with a trip to Old Trafford.
Manchester United still need to get the better of Cambridge United in a a replay following a 0-0 draw on Friday in order to join the Bantams in the hat.
If I had my choice, I'd like to play Manchester United.
The experience we had yesterday going to Old Trafford was just terrific. If Manchester United can manage to beat Cambridge it would be great to go to Old Trafford.
But whoever we get, it's great to be in the fifth round. We're enjoying being in the limelight.
A dressing room visit from Jose Mourinho was the "icing on the cake" for Bradford goalkeeper Ben Williams after the Bantams' astounding 4-2 comeback win over Chelsea in the FA Cup on Saturday.
League One Bradford were 2-0 down in their fourth-round tie at Stamford Bridge after goals from Gary Cahill and Ramires for the Premier League leaders.
However, a strike before half-time from Jon Stead followed by goals from Filipe Morais and Andy Halliday put the Bantams ahead before Mark Yeates confirmed their progress in injury time.
Mourinho admitted he was "ashamed" and "embarrassed" after the defeat, but the Portuguese was keen to congratulate the Bradford players after the match.
And for Williams, the humility shown by Mourinho added to what was already an incredible day.
"It was fantastic. The scenes, the look on everyone's faces was amazing, something which will live long in the memory,"
"The fans were amazing and after that Mourinho came into the dressing room and shook everyone's hand and congratulated us, that was a surreal moment as well.
"In he walked and the celebrations stopped. This figure waltzed into the dressing room. He was very classy, very humble, very respectful - it was the icing on the cake."
Bradford looked to be heading out of the competition after conceding two first-half goals, but Williams believes the combination of sticking to their game plan and perhaps some complacency from Chelsea contributed to their comeback.
"Once you go 2-0 down it was a case of seeing it out and sticking to the game plan and the goal before half-time gave us a real lift," he said.
"We had a way of playing and the boys executed that fantastically well.
"At 2-0 at home everything is in Chelsea's favour, maybe there was an element of taking their foot off the gas.
"It's the FA Cup - it's a fantastic leveller and you have always got a chance if you remain in the game. Take nothing away from Bradford, the goals we scored were fantastic."
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