A dry opening half resulted in fewer scoring chances, with each side reduced to shots from range, with Marcus Rashford firing at John Ruddy on 14 minutes.
As the half drew to a close, Diogo Jota latched onto a through ball from Ruben Nevas but his close-range shot was turned away by Sergio Romero.
United were under increasing pressure in the second half as Wolves continued probing, with Raul Jimenez’s header pushed over the top of the crossbar by Romero on 55 minutes.
While Wolves looked to subside, Jimenez wiggled through the defence to break the deadlock with a shot on the turn after 70 minutes.
Just as a come back beckoned, Jota dispossessed Luke Shaw to go through on goal and beat Romero with a near post finish on 76 minutes.
The Red Devils were handed a red card following a reckless challenge by Victor Lindelof on 83 minutes, but the Video Assistant Referee overturned decision and the defender was only handed a yellow card.
United grabbed a consolation in added time, as Rashford found the bottom corner from poor defending, but Wolves hung on to secure a semi-final berth.