Scott McTominay had put United into an early lead with his first goal for the club, but Diogo Jota's low-range strike got Wolves back on level terms before Ashley Young was dismissed before the hour mark.
United matched Wolves during the closing quarter of the match, but Chris Smalling put through his own net with 13 minutes left as United missed the chance to move into third place.
Ole Gunnar Solskjaer criticised his players for the tempo of their play when losing to Wolves in the FA Cup quarter-finals last month, but United could not be accused of playing at a slow pace during the opening 15 minutes.
Jesse Lingard forced Rui Patricio into a save within the first 15 seconds after a mazy run from kickoff, while the Portuguese stopper also had to react well to keep out Romelu Lukaku's header from point-blank range.
Wolves are seeing plenty of the ball but United deservedly went ahead when McTominay was able to find the bottom corner of the net with a low drive from 20 yards out.
Lingard should have doubled United's lead from close range only for Patricio to tip his header away from goal, but Wolves were able to appreciate the reprieve and they promptly capitalised through a mistake from Fred.
The Brazilian was dispossessed around 35 yards from goal after receiving a pass from David de Gea, and Raul Jimenez was able to find Jota who confidently fired into the bottom corner from eight yards out.
Wolves found themselves in the ascendancy and Leander Dendoncker squandered a chance on the volley from eight yards out, with the Belgian sending his effort over the crossbar after good work from Matt Doherty.
Ruben Neves also saw a long-range volley claimed by De Gea from just underneath his crossbar, but United should have regained the lead just before the break with Lukaku only dragging a volley wide of the far post after being played in by Paul Pogba.
McTominay nearly added a second shortly after the restart with Patricio again making a save from a close-range header, but United were reduced to 10 men shortly before the hour mark as Young picked up two yellow cards for fouls on Jota.
The sending-off acted as the catalyst for Wolves to press forward in numbers, although the home side soon began to fade as the game entered the final quarter having failed to create a noteworthy opening.
United took the initiative until three substitutions disrupted the flow of the game, and it was Wolves who took full advantage by completing the turnaround with 13 minutes remaining.
Joao Moutinho's cross from the left-hand side caused confusion in the penalty area, and pressure from Dendoncker forced Smalling to divert the ball past De Gea from close range.
Wolves created chances to extend their lead, with only a save from De Gea preventing Smalling from putting through his own net for a second time and Ivan Cavaleiro hitting the crossbar from 12 yards.
However, the home side were able to survive late pressure from United to return to winning ways ahead of Sunday's FA Cup semi-final with Watford at Wembley Stadium.
WOLVERHAMPTON WANDERERS (3-5-2): Rui Patricio; Bennett, Coady, Boly; Doherty, Dendoncker, Moutinho, Neves (Saiss 84'), Vinagre (Jonny 76'); Jimenez, Jota (Cavaleiro 73')
MANCHESTER UNITED (4-3-3): De Gea; Young, Smalling, Lindelof, Shaw; Fred (Jones 65'), McTominay, Pogba; Dalot (Pereira 84'), Lukaku (Martial 73'), Lingard
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