A collision with Mateo Kovacic on 68 minutes forced Bailly off with a knee injury.
"He will need a scan tomorrow but I don't think it is the worst knee injury you can have," Solskjaer told manutd.com. “He will be out for the rest of the season, I don't think we will see him again before next season.
"But he is a warrior, he played well today, he was calm, he was composed and it was just unfortunate that he had to come off."
Marcus Rashford also collected a shoulder setback.
Solskjaer added: “Rashy's struggled with his shoulder and he came through an hour, but couldn't do more."