It was the first defeat for the hosts since interim manager Ralf Rangnick took charge at Old Trafford late last year and the underwhelming performance shows signs that not all is well at the club.
The German coach has attempted to overhaul certain aspects of United's game but, yet again, the formation seemed to lack structure and players looked short of confidence.
Wolves, meanwhile, were worthy winners after dominating large portions of the match.
Their midfield simply outclassed United's, and the visitors could have scored even more if it hadn't been for a string of saves from United goalkeeper David de Gea.
Lack of composure
The goal finally came in the 82nd minute of the match, with veteran midfielder João Moutinho firing low into the bottom corner.
It secured Wolves' first win at Old Trafford since 1980 and moved the side up to eighth in the table -- just one place behind Manchester United.
"We didn't play well at all, individually or collectively. In the first half, we had big problems to keep them away from our goal," Rangnick said after the match.
"The game showed we still have a lot of work to do. We had too many unforced errors.
"I don't want to speak about individual performances of players. It's an issue of the whole team. It doesn't make sense."
After winning its last game comfortably at home to Burnley, fans at Old Trafford would have expected another strong performance on Monday.
But from the first whistle, United's stars seemed to lack composure as they failed to sustain any kind of pressure on Wolves' goal.
The second half looked slightly better after the introduction of playmaker Bruno Fernandes. The Portuguese star rattled the crossbar shortly after coming on as a substitute before Cristiano Ronaldo's goal was rightly disallowed for offside.
Fernandes also had a free-kick saved in the final seconds of the game.
'I knew it would be difficult'
One potential positive for United is the return of Phil Jones. The center-back has been out with injuries for the last two years but was handed a start with Untied suffering an injury crisis in defense.
The 29-year-old delivered a more assured performance than many of his teammates who were jeered by some members of the crowd at full time.
"I knew it would be difficult," Rangnick said in a brutal assessment of the job at Manchester United.
"That's the reason they brought me. I knew it wouldn't be easy to find a balance between offensive and defense.
"Today, we showed similar problems the team had three or four weeks ago before I arrived. It is not an easy job to develop a DNA with and without the ball.
"The game today showed there are still quite a few steps to go."
United faces Aston Villa in the FA Cup on Monday before playing them again in the league on January 15.
The Red Devils are currently four points off the Champions League qualification places.