Mourinho's side never looked to be in complete control at Ashton Gate, even if they hit the frame of the goal through Zlatan Ibrahimovic and Marcus Rashford in the first half.
City, who played with greater urgency for most of the match, broke the deadlock early in the second half through a stunning Joe Bryan goal, but Ibrahimovic levelled soon after from a free-kick.
And just when the game looked destined to go to extra-time, Korey Smith scored in the last minute to send Championship side City into the semi-finals, though Mourinho thinks they can consider themselves fortunate.
"My thoughts are they were a bit lucky," he told Sky Sports. "But they fought a lot to be lucky.
"They won the game in a moment when we have no possibility to react. They won in our best moment of the match when everyone was waiting for our goal.
"I have to say they were lucky. We also hit the post twice.
"But they fought brilliantly and like it was the game of their lives, which it probably was.
"It was a beautiful day for football. I don't want me saying they're lucky to be the main thing
"The intensity they had [was their biggest asset], not just physically, but also mentally. For some of us it was one more day in the office, for some maybe a day at the office that they didn't want.
"For them it was a big, big day. They were a bit lucky and we weren't so bad, like after the goal in second half we were the best team and we tried to win in the 90 minutes.
"They were very lucky in the last second to win. But again, I don't want me saying they're lucky to be the main thing. We know how to lose and win and we say congratulations and wish them well in the semi-finals."
Bristol will face Manchester City in the two-legged semi-final.