Bruno Lage's side were excellent at Old Trafford and they could have put the game to bed in the first half as they were the better side throughout.
It was Wolves' first league win at the Theatre of Dreams since 1980 and they fully deserved it as they did exactly what Rangnick craves his side to do...control the game.
As one Portuguese midfield magician struck the bar in the second half, the other got the winner.
It was Adama Traore's cross that forced Phil Jones to head clear which found Moutinho on the edge of the box. The midfielder took a touch before firing into the bottom left corner.
Story of the game
Wolves had a bright start and enjoyed plenty of possession in United's third, however, it was the hosts who should have taken the lead in the opening 10 minutes.
Mason Greenwood won the ball in his own half and he picked out Jadon Sancho on the far left. Sancho dribbled the ball up to the 18-yard box but his shot was blocked. He had Cristiano Ronaldo to his left and Edinson Cavani to his right, both arguably in better positions.
Wolves came straight back at United through Daniel Podence. His tricky run beat Aaron Wan-Bissaka and his snapshot forced David De Gea to parry wide for a corner. The corner was cleared but came out to Ruben Neves who volleyed from outside the box and De Gea made another good fingertip save.
United struggled to string a sequence of passes together and once more proved their lack of connectivity. Their midfield failed to combine with the forwards and at times it felt as though Wolves were the home side with the way they controlled the match.
Raul Jimenez then headed wide with arguably the best opportunity of the first half when Romain Saiss picked him out at the front post. It came from a shoddy back-pass by Ronaldo yet Wolves failed to capitalise.
Both teams came out for the second half unchanged, but Fred and Bruno Fernandes were warming up from the whistle.
Little had changed before the hour mark as United still looked lethargic and Wolves contained them with ease.
Rangnick made the first move as he subbed Greenwood for Fernandes. Greenwood had created the most for United and the home fans let their interim boss understand their anger as they booed the decision.
Lage then reacted and brought on Adama Traore for Trincao. It had already been an uncomfortable night for United's defence - and many felt it could get even more so with the Spaniard's blistering pace.
United finally got going when Fernandes was introduced. Nemanja Matic found space down the left and zipped a ball across the 18-yard box. The Portuguese midfielder met the ball and hammered it off the crossbar.
Moments later, United won a free-kick on the far right and swung a deep ball in for Ronaldo to head into the back of the net. He was correctly ruled offside, as he failed to time his run.
It was United again who came forward through Ronaldo. He used his pace to get in behind Conor Coady before dragging an effort wide.
Those three attempts lured the United faithful to perk up and started chanting 'attack attack attack attack attack'.
Traore then won Wolves a free-kick right on the edge of the box. Saiss and Moutinho stood over it but it was Saiss who went for it and the ball rocketed off the crossbar.
But it was Wolves who wrote the final chapter as Moutinho perfectly found the bottom left corner of De Gea's net.
Fernandes' late free-kick then forced Jose Sa into a spectacular save to secure the three points and the final whistle followed.
Wolves had scored their first away goal at United from their last three visits and it came at the perfect time.