The Championship outfit booked their place in the semi-finals with a last-minute goal against the Red Devils.
Joe Bryan put Bristol ahead before Zlatan Ibrahimovic equalised with a stunning free-kick.
Korey Smith then sparked jubilation at Ashton Gate by scoring the winning goal in the third minute of added time, setting up a next round showdown with Manchester City.
Conte’s side also had their own late drama as they beat Bournemouth 2-1 in stoppage time.
And the Italian praised Bristol City for overthrowing their fourth premier League opposition this season.
“It’s a great achievement,” Conte said.
“Manchester won the competition last year.
“You must pay attention in every game.
“These cups are very good for these types of surprise.
“I knew the final result in the changing room.
“It’s a surprise but in England it’s difficult to face every team.”
Chelsea drew Arsenal in the other semi-final clash.