Antony had forced the game into extra-time with a late equaliser for 2-2, but the dramatics weren't over as Marcus Rashford and Amad Diallo struck after Harvey Elliott had put the visitors ahead.
Klopp said: "We did not have a good start but we adapted really well and played a top game. At 2-1 up we had so many chances to kill the game off, we were in a rush in some moments and the rest is already history. We were nearly there in extra time and in the end we lose the game. I cannot ask the boys for more.
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"We had an intense season so far, you just have to count the games and today was hard. United took a lot of risks, I absolutely respect that.
"The decision making doesn't get better the longer the game goes on. Congratulations to United for going to the semi-final."