Solskjaer hits back after criticism over Rashford injury
The United and England striker could be out for three months after suffering a back injury in last week's third-round match, which United won 1-0.
Former England and Arsenal striker Ian Wright said this week that Solskjaer, under intense pressure to improve the club's fortunes, prioritised his own needs over those of his player.
United are in fifth place in the Premier League, five points behind fourth-placed Chelsea in the race for the Champions League qualification spots.
But the Norwegian boss insisted that Rashford was fit enough to play in the 1-0 win for United, in which the 22-year-old substitute lasted just 16 minutes.
"I never put myself before the team," Solskjaer said on Tuesday. "I always put the team and the club before anything else.
"Marcus has had minor discomfort in his back since the Burnley game, two weeks before this game (Wolves). We've had scans, we've monitored him, there was no injury there, just complaints."
Solskjaer, whose team host Burnley in the Premier League on Wednesday, said he had spoken to Rashford in the build-up to the match.
"I had a conversation with him two days before and told him I was going to rest him," he said. "He felt fine and was very adamant that if needed, he could do at least half an hour as we wanted to go through in the cup. That shows you Marcus's attitude."
Solskjaer said he did not regret throwing his star striker into the fray.
"That is part of what you do for Man Utd," he said. "Every time you play, you put your body on the line. I've suffered some injuries for the club and that is what you are proud to do when you are a Man Utd player."
Solskjaer's biggest headache now is how to cope without his leading scorer and talisman as he battles to finish in the Premier League's top four.
United's attack looked weak without him in the 2-0 defeat against Liverpool on Sunday and Anthony Martial and Mason Greenwood are the only striking options for the Burnley game.
"That is the million-dollar question for us because Marcus has been fantastic," Solskjaer said.
"Someone has got to step up. If we find something in the transfer market, that might be one solution. But the best solution is the players stepping up and making sure they are ready and taking the chance."