Rashford was introduced in the 64th minute with the game goalless, but only lasted 16 minutes before he went off with a back injury.
The 22-year-old, who has scored 14 Premier League goals this season, is now a doubt to face Premier League leaders Liverpool on Sunday.
"He'll be tested in the next few days - we have to see for Sunday," said United manager Solskjaer.
"I didn't want to play him (against Wolves). I think he got a knock, a knee or something, in his back, but he's been struggling for a little while.
That's why we've kept him away (from starting against Wolves), but we needed the win. He was involved in the goal, so that was a positive, but it backfired.
"He's a quick healer and he'll play through some pain, definitely, if he's capable and it's not a bad, bad injury, but we've seen that he's struggled lately. But that's been more when he's been tired. He wasn't on long enough to be tired so it might be something that just happened there and then.”
United are the only side to have taken points from Liverpool in the league this season as they held them to a 1-1 draw at Old Trafford in October.
Marcus Rashford scored against Liverpool in October
Image credit: Getty Images
But losing Rashford, who scored the opening goal in the draw against Liverpool, would be a big blow to United's chances of a positive result at Anfield.
"He's been absolutely top this season so we'll do anything and everything we can to get him fit for Sunday,” said Solskjaer.
“If he's not, then we'll play without him.
United are in fifth place, five points behind fourth-placed Chelsea and 27 points behind Liverpool, who have not lost a home match since April 2017.
"We have found a nice way of playing against them and have deserved the point," added Solskjaer. "We'll wait and see."