Ange Postecoglou's side were cruising against a lacklustre United with Dominic Solanke's double and a goal for Dejan Kulusevski putting them 3-0 up after 54 minutes on Thursday.
Tottenham players celebrate after Dominic Solanke scored Spurs's third goal. (AP PHOTO)
But the home crowd were left chewing their fingernails as two terrible mistakes by 'keeper Fraser Forster allowed substitutes Joshua Zirkzee and Amad Diallo to give United hope.
There were still 20 minutes left when Diallo made it 3-2 and United were close to equalising on several occasions before Son Heung-min appeared to have sealed it in odd fashion, deceiving United keeper Altay Bayindir directly from a corner.
There was still time for Jonny Evans to head a third for United in stoppage time but a relieved Tottenham hung on.
Tottenham are seeking a first trophy since 2008 when they won the League Cup for the fourth time.
Ange Postecoglou came out on top in a seven-goal thriller at Tottenham Hotspur Stadium in London. (AP PHOTO)
Postecoglou's side let a 2-0 lead slip to lose 4-3 at home to Chelsea in their last home game and for a while it looked as though their innate ability to shoot themselves in the foot was coming back to bite them.
This time, however, the fans went home happy as Postecoglou moved a step closer to making good on his reputation for securing silverware in his second season at clubs.
Tottenham are at full stretch because of injuries but still dominated Ruben Amorim's side which showed five changes to the starting line-up that stunned Manchester City on Sunday.
Solanke put the hosts in front in the 15th minute when Bayindir fumbled a well-hit shot by Pedro and the striker fired in the rebound off the post.
Bruno Fernandes had a couple of efforts and Rasmus Hojlund volleyed wide but Tottenham were comfortable and in cruise control straight after the interval when Lisandro Martinez could only turn a James Maddison cut-back into the path of Kulusevski who made no mistake.
Solanke was allowed to dribble across the area before sliding a shot inside the post in the 54th minute and the tie looked over as a contest.
But Forster then played a casual pass out of his area and United seized the ball for Zirkzee to tap in.
Amad Diallo (c) retrieves the ball after scoring a quick second goal for the Red Devils. (AP PHOTO)
Forster then compounded that error by taking an age to make a clearance and Diallo slid in to charge down his kick and divert the ball into the net.
Tottenham were a bundle of nerves as United swept forward but captain Son restored calm with an inswinging corner that went over the head of Bayindir to ripple the net.
Substitute Evans' riposte came too late to save United.
Meanwhile, the north-London side has been rewarded with a two-legged clash against Arne Slot's Premier League table toppers Liverpool.
The opening leg will be at the Tottenham Hotspur Stadium, with fellow semi-finalists Arsenal also handed home advantage for their first leg, in the week of January 6, against Newcastle.
The return fixtures will be played on the week commencing February 3. The final will take place at Wembley on March 16.
Semi-final draw:
Arsenal vs Newcastle (first leg)
Tottenham vs Liverpool (first leg)
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