Defeat leaves United fifth, outside of the top four on goal difference having played a game more than the side's around them, and manager Solskjaer admits the club face a hard task to qualify for next season's Champions League.
The result also gave it plenty to ponder before next week's Champions League quarterfinal first leg against Barcelona.
Wolves created chances to extend their lead, with only a save from De Gea preventing Smalling from putting through his own net for a second time and Ivan Cavaleiro hitting the crossbar from 12 yards.
"It was a very good start, we should have been 3-0 up and we created our own downfall really", said Solskjaer.
McTominay scored United's goal, it was his first for the club.
"I would believe so, that's our aim as a club to be achieving what we have done in the past and that is the expectations here and of course we are hoping to build a team strong enough worthy of the history of the club", the Norwegian said recently.
However, he hailed his charges for a good show up-to the 75th minute when the conceded the second goal.
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"To do it twice against such an incredible team. So I was very, very pleased with him".
"United began the game very well".
10-man United failed to get themselves back into the game and could now see themselves fall to sixth in the Premier League, behind rivals Chelsea. "There's no secret. Work".
Wolves impressive return to the Premier League continues as they solidified their hold on seventh place with a haul of 47 points through 32 matches.
United had the chance to move third in the Premier League but endured a miserable night.
Timo Werner had given the visitors the lead in the 74th minute.
RB Leipzig recovered from conceding a stoppage-time equalizer to score a victor in the last minute of extra time as it beat Augsburg 2-1 Tuesday to book its place in the German Cup semifinals.