But Diogo Jota levelled after Fred gave away possession and United captain Ashley Young was sent off after being booked twice for fouls on the Wolves striker to become the 100th dismissal of Mike Dean's Premier League career.
United had appeared in control despite Wolves having a man advantage and did not concede a single effort on David de Gea's goal until a Joao Moutinho cross was turned into his own net by Chris Smalling.
A run of three defeats in four games in all competitions has seen United lose ground at a crucial time of the season but Solskjaer was still satisfied with the way his side performed.
"[It was] one of those days that we have done more than enough to score two or three goals. We had great chances and some fantastic saves from the keeper," Solskjaer said to Sky Sports . "Of course they deserve credit, they scored and pounced on our mistakes.
"I thought the lads played well, created chances with 10 men as well. They scored from their first chance in 25 minutes after the sending off. The boys did what they should but we conceded from sloppy mistakes and didn't score.
"It happens. It's just a couple of bad decisions but that's football, we trust them to play and they played some great football but the downside is you take some risks.
"I thought we did well at the start of the second half, Scotty [McTominay] had a fantastic chance and a great save. I think he made three or four fantastic saves. We did enough to win the game but you don't always get what you deserve."
How much for these two? pic.twitter.com/U9wv9YCarY— Wolves (@Wolves) April 2, 2019
Wolves have won more points against 'big six' sides in the Premier League this season than any of the other 14 teams, taking 13 from games against United, Liverpool, Manchester City, Chelsea, Arsenal and Tottenham.
It was a second win in three weeks over United at Molineux for Nuno Espirito Santo's men and he hailed the reaction of the FA Cup semi-finalists to falling behind.
"Good performance, in the beginning they started better and controlled the game," Nuno said. "We had trouble with our shape and they had possession and scored.
"That moment we suffered and took time to react, but we had a fantastic reaction and managed the game better.
"After the red card our task is not so difficult, of course it is always dangerous against a difficult team, but it makes me proud against such a good team.
"The reaction and the way they reacted to conceding a goal - we know we can concede again and it's game over but we stayed in the game and stayed calm and organised. The way we went forward was amazing."