The home side were cruising at 3-0 in the 54th minute of a captivating tie in London 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 fourth as Son Heung-min found the net from a corner and they held on despite a late Jonny Evans goal.
Tottenham will face Premier League leaders Liverpool in the two-legged semi-finals, with Arsenal taking on Newcastle.
Spurs boss Ange Postecoglou said his injury-hit team inflicted pain on themselves but gave a passionate defence of his gung-ho approach.
“It should have been a lot more comfortable there, but having said that, I still can’t get away from the fact that this group of players have done an unbelievable job at the moment to get us through this,” he said.
The Australian added: “I know people will say, “Well, don’t take any risks’ and all that sort of stuff, but we wouldn’t be scoring four goals and five goals.
“It’s the hardest thing in the game to do, score goals, and we’re doing it with bare bones.”
Tottenham, 10th in the Premier League, have not won silverware for 16 years, when they lifted the League Cup under Juande Ramos.
But they are now just one stage away from a Wembley final in March, with Postecoglou still on track to deliver on his bold statement that he always wins a trophy in his second season at a club.
New United boss Ruben Amorim is 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 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.
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