After a scoreless first half, Ollie Watkins opened the scoring straight away in the second half.
But the joy was short-lived and Anthony Martial headed home the equalizer just seconds after the restart.
In the 61st minute, Villa took the lead again after Leon Bailey headed the ball down at the feet of Diogo Dalot.
The defender unfortunately angled the ball into his own goal cage.
Marcus Rashford equalized and in the 78th minute came the mistake that decided the decision.
Robin Olsen had the ball at his feet, but hit a weak pass straight into the arms of Alejandro Garnacho.
He in turn found Bruno Fernandes who made it 3-2, while a distressed Olsen sat on the ground clutching his forehead.
Scott McTominay settled the final score 4-2 in stoppage time, and it's done in the League Cup for Aston Villa while Manchester United are through to the fourth round which will be played on December 20-22.
Ludwig Augustinsson played 80 minutes for Aston Villa while Anthony Elanga was substituted in the 62nd minute for United and Victor Nilsson Lindelöf played 87 minutes.