Korey Smith's stoppage-time winner secured a place in the last four for the hosts in a second defeat in four games for Jose Mourinho's men.
Juan Mata was an unused sub at Ashton Gate and was disappointed with the result, but he said the club must move on if they are to achieve success this season.
"When you play for a club like Manchester United every defeat is a massive blow for everyone," said Mata. "For the fans, for you but especially for us. We suffer when we don't win because there's such a high standard in this football club to try and win every game."
"But we need to understand that defeat is part of sport and part of football and it can happen so it's not a tragedy," he added. "It's not good at all, but we need to move on and accept it as a part of football and keep moving. Obvously we would like to win every single game and we would like to try to do so until the end of the season."