Rashford lasted less than 15 minutes on the field due to a back injury and is now a doubt for the trip to runaway leaders Liverpool in the Premier League on Sunday.
The brief cameo from United's top goalscorer coincided with the only goal of the game as Juan Mata's clinical finish ensured United win over Wolves and now will travel to either Watford or Tranmere on January 26th.
"That backfired, but you've got to go for them," Solskjaer told BT Sport.
"He's got a knock and couldn't run. Let's test him for Sunday and see how he goes. He's had a few knocks with his back lately and he got another one".
Wolves put United out of the FA Cup last season and were unbeaten in five previous meetings between the sides since returning to the top flight in 2018.