The Championship team beat Wolves 3-2 with two injury-time goals at Molineux on Saturday, while United overcame Liverpool with a staggering 4-3 extra-time win on Sunday.
Manchester City will take on Chelsea in the other semi-final as they attempt to retain their trophy.
Mark Robins, Coventry manager, will be particularly enthused by his side’s draw after appearing 61 times for United as a player.
Indeed, it was Robins’ FA Cup goal against Nottingham Forest in 1990 that is considered to have saved Alex Ferguson’s job as manager at Old Trafford after just over three years at the time.
It is the Midlands club’s best FA Cup run since they won the competition in 1987.
Their opponents United, will be aiming to make back-to-back FA Cup finals for the first time since 2005.
Man City, meanwhile, must beat Chelsea if they are to become the first side to defend the trophy since Arsenal in 2015.
Pep Guardiola’s team have failed to beat the Blues in either of the two league meetings between both sides this season.
Mauricio Pochettino’s side have already lost the Carabao Cup final this term but overturning City to reach another showdown would be a feather in their cap.
The ties will be played at the Wembley National Stadium, London across the weekend of Saturday, April 20 and the winners of the two games will meet in the final on Saturday, May 25.
