The Reds had a two-goal lead at half-time, after strikes from Manuel Ugarte and Diogo Dalot, but Corentin Tolisso and Nicolas Tagliafico struck back to take proceedings beyond the 90.
United seemed to have the upper hand when Tolisso was shown a second booking and sent off, but the visitors added further goals through Rayan Cherki and an Alexandre Lacazette penalty.
But, in the second part of extra-time, Ruben Amorim's side achieved one of the all-time great comebacks, as a Bruno Fernandes spot-kick was followed by goals from Kobbie Mainoo and Harry Maguire to send United into the semi-finals.