Alli's early finish had given Mauricio Pochettino's side the lead, although Tottenham were unable to prevent United from clawing their way back into the game through Alexis Sanchez, before Ander Herrera sent Jose Mourinho's men into the final with an adroit finish in the second half.
"We let ourselves down, it's very disappointing. We have to try and pick ourselves up and finish the season strongly," Alli told BBC Sport after the defeat.
"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."
Defeat for Spurs has ended the club's final chance of claiming silverware this term.