United lacked spark throughout and were overcome by an eager Wolves side with two goals in the space of six minutes in the second half from Raul Jimenez and Diogo Jota.
Marcus Rashford’s injury-time effort was a mere consolation.
Solskjaer had presided over a run of just two defeats in 18 games since replacing Jose Mourinho in December, progressing to the quarter-finals of the Champions League and the last eight of the FA Cup.
But now they have suffered two losses in a row after their Champions League exertions at Paris Saint-Germain.
United were favourites but the Midlanders’ form against the top six that has carried them to seventh position in the Premier League gave home fans hope.
The early signs were that United, not Wolves, were going to live up to the hype of the warm-up, dominating early possession.
The visitors created the first opening when Paul Pogba gained a yard of space in the box and curled wide.
Goalkeeper John Ruddy was happy to let the ball drift away but was forced to make a falling save to his left soon afterwards when Rashford let fly from 30 yards.
Wolves’ first threat came in the 28th minute when Ruben Neves’s low drive was well held by the United goalkeeper Sergio Romero and it was swiftly followed by a better one, Jota failing to connect properly with a volley from Matt Doherty’s centre.
United players surrounded referee Martin Atkinson screaming for a penalty in the 37th minute after Diogo Dalot’s hooked shot hit the elbow of Jonny as he turned his back on the ball. But Atkinson was unmoved.
The best chance of the half fell to Wolves, two minutes before the break. Neves sent Jota galloping clear for a one-on-one with Romero, but the United goalkeeper beat his curling shot away.
United had the first opening of the second half when Rashford got away from Conor Coady on the right side of the box but could only whistle a fierce angled drive a few feet wide.
Wolves replied with a rising, diagonal effort from Jimenez that looped comfortably over the bar.
Nuno Espirito Santo’s side were gaining in confidence however and created two chances that required some superb goalkeeping from Romero.
First he reacted sharply to tip over a bullet header from Jimenez from Joao Moutinho’s corner.
The Argentine keeper then palmed over a rising chip from Moutinho from 25 yards as Wolves got on top.
Wolves scored the goal they had been threatening in the 70th minute.
The chance seemed to be lost when Jimenez was forced backwards with his back to goal after being crowded out following a superb run by Moutinho, who breezed through three challenges cutting in from the left.
But the Mexican swivelled 180 degrees, after getting a lucky bounce off Pogba and drilled a low shot past Romero.
Six minutes later Wolves doubled their lead.
Neves sent Jota racing clear on the counter as Wolves broke and the Portuguese left Luke Shaw in a heap before firing an unstoppable left-foot drive inside Romero’s near post.
High drama followed when Atkinson showed Victor Lindelof a red card for lunging in on Jota in front of the Wolves dugout.
VAR was called and deemed it a booking, allowing the Swedish defender a reprieve.
Rashford pulled one back with a shot on the turn in the fifth minute of the five added on, but it was too little, too late.