Bryan gave City a 51st-minute lead but Zlatan Ibrahimovic equalised shortly afterwards.
Then, just when extra-time beckoned, Korey Smith was set free by Matty Taylor and smashed home the stoppage-time winner.
Bryan said: “We don’t go into any game this season we think we’re going to lose. I think we’re full of confidence and belief in yourself. I think we showed that tonight.”
But the 24-year-old full-back can remember little about his second goal in three games.
On trying to recall the goal, he added: “I don’t know. Honestly. It just broke and I hit it and it’s gone in. I can’t remember much until I see it again.
“Since we’ve (current crop of players) been playing here it’s definitely the best night we’ve had.”
Smith was confident he could hurt United if he was just given the service.
He said: “A few times I’d made runs in behind and I wasn’t being tracked so I said at half-time to the boys, ‘if I make that run look for me and I’ll try and get that shot off’.
“Luckily Tayls (Taylor) made a great ball and it’s gone in.
“I joined the club when we were in League One. We had three stands and obviously we had a great season. And the next year we were in and around the bottom of the (Championship) table.
“This year we’ve been absolutely brilliant. The younger players are developing and people are coming into their prime. The stadium now is absolutely amazing. It’s nice to give the fans a season they’re enjoying so far.
“To be in the draw with three of the best clubs in the country — everybody would have expected the top four teams in the country to be in the draw for the semi-final — it’s just been unbelievable. It’s just absolutely amazing today.”
Bristol manager Lee Johnson hailed last night’s win an incredible achievement.
He said: “It’s an overwhelming feeling of pride, to beat Manchester United is incredible. They are a special group. It has been a long time since we had a result like that at Ashton Gate.
“The lads didn’t want to go into extra time and the scenes that winner set off were unbelievable. My dad always says he’s the most successful Bristol City manager so I have maybe pushed him close tonight! I don’t think many fans will be turning up for work tomorrow, they’ll be on the cider.”
Manchester United manager Jose Mourinho said Bristol City got lucky.
“They were a bit lucky but they fought a lot to be lucky,” he said.
“Everyone was waiting for our goal so they were lucky. We hit the post twice.
“But they played brilliantly, they fought like it was the game of their lives which probably it was. A beautiful day for football. A team from the lower division won. A big day for them.
“In the first half we lacked the intensity that they had. Physically and mentally. It was one more day in the office for my players, a day that some of them didn’t even want to come to the office.”
Steele, Wright, Flint, Baker, Magnusson (Taylor 69), Brownhill (Eliasson 74), Pack,Smith, Bryan, Paterson, Reid.
Romero, Darmian (Smalling 90), Lindelof, Rojo, Shaw, McTominay, Blind (Lukaku 61), Pogba, Rashford, Ibrahimovic (Mkhitaryan 69), Martial.