Rashford made a lung-busting run to prevent a City counter-attack in the second half and appeared to hurt his left ankle in the process.
The Englishman tried to continue but was unable to move freely, prompting Solskjaer to substitute him for Mason Greenwood in the 73rd minute.
Rashford’s potential injury comes as a blow to Solskjaer, who confirmed the 23-year-old will be sent for a scan.
Asked about the severity of Rashford’s injury, the Red Devils boss said: “I don’t know. It’s something with his ankle.
“When he sprinted 70 yards to save a goal, it showed the attitude of the boy.
“I think he went over on his ankle but we need to do a scan, I’ve not had time to speak to the doctor yet.”