The draw took place directly after United's epic 4-3 win over Liverpool, which saw Erik ten Hag's side score late in extra time to stage a fightback.
Their reward is a meeting with a Coventry side led by former United man Mark Robins, who famously scored an FA Cup-winning goal for the Red Devils against Nottingham Forest in 1990.
The Sky Blues earned a last-gasp quarter-final comeback of their own on Saturday, winning 3-2 on their travels against Wolves.
The other semi-final sees reigning FA Cup champions City, who brushed off Newcastle in the last eight, face a Chelsea side who suffered a major scare at home to Championship leaders Leicester.
The Blues won 4-2 in a frenetic encounter at Stamford Bridge which saw the home crowd briefly turn on manager Mauricio Pochettino for the team's second-half display.
The ties will be played across the weekend of April 20th at Wembley Stadium.
