United were poor for much of the first half but still had a penalty overturned and a goal ruled out by VAR.
United dominated the second half but could not pass goalkeeper Karl-Johan Johnsson, with Fernandes and Greenwood also each hitting the post and Rashford having another effort disallowed.
Anthony Martial was fouled inside the box and Frenandes step up to take the penalty and give United the lead in the 95 minute.
Karl-Johan Johnsson was magnificent between the posts for FC Copenhagen – ending the night with 13 saves.
United will now face either Premier League rivals Wolves or Europa League specialists Sevilla in the last four in Dusseldorf next Monday night.