However a goalkeeping mistake from Gianluigi Buffon helped United score the second goal through Lukaku and with three minutes to stoppage time, United were handed a penalty following a VAR decision and Marcus Rashford held his nerves to send them through to the last eight.
United had to survive until the game's 100th minute before joyously celebrating a famous Champions League victory in front of their traveling fans.
Meanwhile, Jose Mourinho is the first choice to take over at Real Madrid and has already been approached by the club, according to former president Ramon Calderon.
Mourinho, who left Real in 2013, said last week he was flattered to be linked with a return to Real, and appears to be a popular choice among fans, who were chanting his name after Tuesday's humbling 4-1 defeat by Ajax.
Calderon says Real's president Florentino Perez needs a bombshell "to keep calm among the supporters and is set to call on Mourinho, the "only coach he respects.