Amad Diallo scored the winning goal in the 120th minute with his first goal of the season.
Alexis Mac Allister equalised after his close range shot deflected off Kobbie Mainoo, a few minutes later Mohammed Salah was presented with an easy opportunity which he finished with aplomb to turn tge game around for Liverpool.
Substitute Antony restored parity for Manchester United to send the game into extra time.
Harvey Elliott restored Liverpool’s lead with a deflected shot in the first half of extra-time.
He was later sent off for taking his shirt off while celebrating the goal after receiving a yellow card before popping up with the winning goal.
Ten Hag has hailed his players for their impressive display to book another Wembley clash.
He said: “To go to the final we have to win against Coventry, we will have to play at our best. We saw that Coventry have great spirit, a great mentality. We will prepare our best to play in the final.
“The first 30 minutes was the best of our whole season, we were acting as a team but then we got gaps between the lines and you can’t allow this against one of the best teams in Europe. They outplayed us.
“Then we made changes, we took risks and the players were wonderful. The attitude was great, a strong belief to win this game.
“Rashford is resilient, he keeps going
Manchester United will take on Coventry City in the semi final, while Chelsea will battle it out for a place in the final against Manchester City.