Spurs deservedly took the lead 11 minutes into Saturday's encounter at Wembley when Alli pounced to turn home a cross from Christian Eriksen.
However, goals from Alexis Sanchez and Ander Herrera turned things around in London and sent United into the final, with Spurs now having lost in all eight of their most recent appearances in the last four.
Mauricio Pochettino's side suffered a similar collapse at home to Juventus in the Champions League last-16 second leg, with Giorgio Chiellini rather scathingly describing the result as "the history of Tottenham", adding "they always miss something at the end".
With their hopes of a first trophy since 2008 over for another season, Alli admits something has to change.
"We let ourselves down, it's very disappointing," he told BBC Sport. "We have to try and pick ourselves up and finish the season strongly.
"You can't go 1-0 up and then 2-1 down against a team like Manchester United. We can't put ourselves in that situation.
"We were dominant from the first minute and it only looked to be going one way.
"Everyone talks. We want to win trophies, we have the staff that want to win. We can't keep doing this. We can't throw it away. We have got to improve."