The home side were cruising at 3-0 in the 54th minute after two goals from Dominic Solanke and one from Dejan Kulusevski.
But Forster's mistakes allowed substitutes Joshua Zirkzee and Amad Diallo to pull the score back to 3-2 with 20 minutes to go.
It was one-way traffic after that, with United totally on top, but Spurs scored a late fourth as Son Heung-min found the net from a corner and they held on despite a late Jonny Evans goal.
Tottenham have not won silverware since winning the League Cup in 2008, under the management of Juande Ramos.
But they are now just one win away from a Wembley final in March, with Ange Postecoglou still on track to deliver on his bold statement that he always wins a trophy in his second season at a club.
Despite the drama, the defeat will send United boss Ruben Amorim back down to earth with a bump after his side's dramatic 2-1 Premier League win at Manchester City on Sunday.
Amorim made five changes from the team that beat City while injury-hit Spurs had six academy players on their bench.
Marcus Rashford was left out of United's squad for the second consecutive game but Alejandro Garnacho, also absent on Sunday, was on the bench and came on as a second-half substitute.
Solanke double
Tottenham opened the scoring in the 15th minute when Solanke reacted quickest after United goalkeeper Altay Bayindir parried a shot from distance into his path.
Bayindir, in for Andre Onana, should have pushed the ball wider and the United defence were sluggish as Solanke took full advantage, finishing neatly off the near post.
Former Spurs player Christian Eriksen, now at United, had a couple of efforts that were easily handled by the home defence and Bruno Fernandes was impressive on the left side of the United attack.
At the other end, Bayindir was forced to make a smart save at his near post to deny Kulusevski.
Kulusevski doubled Tottenham's lead moments after the restart when Lisandro Martinez failed to clear a James Maddison cut-back.
They were 3-0 up when Solanke ran onto a long ball over the top, drifting inside Martinez and Evans and finding the back of the net.
But United got a foothold in the game just after the hour mark following some chaotic goalkeeping from Forster.
The goalkeeper, being closed down by Zirkzee, passed the ball straight to Fernandes, who found Zirkzee and the Dutchman slotted the ball into an empty net.
Suddenly the match changed complexion, with nerves gripping Spurs.
Minutes later United scored again when Forster dithered after receiving a backpass.
Diallo closed him down in a flash and slid in as Forster tried to clear, with the ball hitting Diallo and ending up in the net.
United continued to pour forward but could not find another goal and Son gave Tottenham a cushion in the 88th minute.
Even then United had time to find another, with Evans heading in from a corner, but they ran out of time.
Former Celtic boss Postecoglou made himself a hostage to fortune earlier this season when he said he always wins silverware in his second season.
Spurs have flattered to deceive in the Premier league, lying 10th in the table with seven wins and seven defeats from their 16 matches.
The other League Cup semi-finalists are Liverpool, Arsenal and Newcastle.
© 2024 AFP