The high-flying Championship side upset the odds on Wednesday night as Korey Smith's last-gasp goal stunned United at Ashton Gate.
That was the fourth top-flight side the Robins have knocked out of this season's competition and they will get the chance to collect another scalp when they take on the Premier League leaders.
In the other last-four clash, Chelsea will take on Arsenal in a repeat of last season's FA Cup final.
Ties will be played over two legs in the weeks commencing January 8 and January 22.
Bristol City, who are third in the
Championship, had already beaten Watford, Stoke and Crystal Palace before seeing off United to set up a clash with Pep Guardiola's side.
The Robins will travel to the Etihad Stadium first before hosting in-form City at Ashton
Gate later that month.
"It is brilliant," City boss Lee Johnson told Sky Sports.
"On to another big one against Manchester City over two legs. It is a chance for the players to test themselves against the best
players." Manchester City required penalties to see off Leicester on Tuesday night after Jamie Vardy's penalty in added-time had secured a 1-1 draw at the end of 90 minutes.
Chelsea will host Arsenal at Stamford Bridge in the first leg of their semi-final after leaving it late to beat Bournemouth.
The Cherries thought they had taken the game into extra-time when Dan Gosling scored in the 90th minute, cancelling out
Willian's 13th-minute opener, only for Chelsea to go up the other end and score through Alvaro Morata.
Arsenal brushed aside London rivals West Ham on Tuesday night, winning 1-0 courtesy of Danny Welbeck's first-half strike.