Instead of putting pressure on their top-four chasing rivals, Man United endured their sixth Premier League defeat of the season.
The Red Devils produced just two shots on target and spent a lot of time defending as Wolves took the game to Rangnick's men.
With eight minutes remaining, Bruno Lage's side produced the decisive blow as Joao Moutinho's shot from outside the box landed in the bottom corner.
The goal arrived on the back of a clearing header from Phil Jones as the centre-back attempted to clear Adama Traore's cross. The 29-year-old was making his first United appearance in 708 days.
Rangnick underlined his frustrations in a post-match interview with Sky Sports saying: "We didn't play well at all, neither individually nor collectively. In the first half, we had big problems to keep them away from our box and from our goal.
"Second half, we changed to a different formation with a back three and we had more control in the second half of the game, had then a spell of 15 minutes where we could have scored, hit the crossbar.
"But the goal that we conceded again showed… We had enough players in the box, but we allowed (Adama) Traore, back to goal, to turn and gain speed and bring in the cross.
"The cross had already been defended by Phil Jones, but again the goal that we conceded was one of too many this season where Moutinho could just shoot completely unmarked with no problems, no pressure at all and therefore we are very disappointed about the result, but also about parts of our performance."