Joe Bryan’s opener was cancelled out by Zlatan Ibrahimovic’s fierce free-kick, before Korey Smith netted the winner in the dying seconds of the game.
Bristol City will now face Manchester City in the final four.
“They (Bristol City) were a bit lucky. But they fought a lot to be a bit lucky. They won the game when everyone was waiting for our goal.
“It’s a beautiful day for a football. The team from the lower division won. I don’t want people to think me saying they are lucky to be the main thing. I am happy for them.
“In the second-half after the goal we were the best team, we reacted well. For them it’s a big night, in this club we know how to win and we know how to lose. We wish them luck in the semi-final.
“It was another day in the office and for some of our players they probably didn’t even want to come into the office today,” Mourinho told Sky.