The Robins outplayed the holders for long periods and took the lead after half time when Joe Bryan's brilliant drive beat Sergio Romero.
Zlatan Ibrahimovic, on his first start in eight months after a long-term knee injury, soon equalised with a fine free-kick as United ratcheted up the pressure.
Mourinho threw on Romelu Lukaku and Henrikh Mkhitaryan in search of a late winner in normal time but it was the Championship promotion contenders who produced the sucker-punch.
Korey Smith slid past Romero in the third minute of added-on time to spark scenes of wild jubilation on the pitch, on the touchline and in the stands as the Robins celebrated their first win over United since 1978.
A pitch-invasion at full-time underlined the significance of the victory for Bristol City and manager Lee Johnson, whose side have been paired with Premier League leaders Manchester City in the last four. Chelsea and Arsenal will contest the other semi-final.
But for Mourinho he is left to reflect on a United display in which he said several players "didn't want to come to the office".
"In the first half we lacked the intensity they had," Mourinho told Sky Sports. "That is not just physical, I say also mentally. For some of us it was just another day in the office and for some of them they don't even want to come to the office.
"For the Bristol boys it was a big, big day for them. You could see that. We hit the post twice, the Marcus Rashford shot hits the inside the post and comes out so they were a bit lucky. We were not so bad as that."
Ibrahimovic saw a first half volley deflect onto the bar; Rashford struck the inside of the post and Lukaku was three times brilliantly denied by Luke Steele - the former United 'keeper - as the Old Trafford club piled on the pressure.
And such was the volume of chances that United created, Mourinho couldn't help but suggest that Bristol City were lucky to prevail.
"They were a bit lucky," the four-time winner of the competition added. "But they fought a lot to be lucky. They win the game in a moment when we have no possibilities to react.
"They won the game in our best moment of the game, they win the game in the moment when everyone was waiting for our goal. I have to say they were a bit lucky. I have to say we hit the post twice before they scored the first goal.
"They play brilliantly, they fought like this was the game of their lives which probably it was. It is nice for it to be like that. It is a beautiful day for football because a team from a lower division won and got into a semi-final.
"It is a big day for them and a big day for their supporters. I congratulate to them I don't want the fact that I say the are lucky to be the pin thing of it. The man thing of it is they won and they worked hard for it.
"In the second half after their goals we react well. We were the best team. They tried to win the game in the 90 minutes and they were very lucky to have the goal in the last second. I don't want the fact that I use the word lucky to be the most important thing.
"A Championship team beat a team like Manchester United, a big night for them. At this club we know how to win, we know how to lose so we say congratulations to them and good luck for the semi-final."