The talismanic striker came off the bench for the final half an hour at Old Trafford last night, having replaced Dan James.
The Red Devils went on to beat Wolves 1-0 in their third round replay thanks to a neatly taken goal from Juan Mata.
Rashford, though, failed to finish the game as he was forced off with a back problem after colliding with Matt Doherty.
The England international star was in serious discomfort as he received treatment before trudging off down the tunnel.
Ole Gunnar Solskjaer admitted after the victory that his decision to bring Rashford on backfired.
United now face a tense wait to see if their 22-year-old goalscorer will be fit to face arch-rivals and Premier League leaders Liverpool on Sunday at Anfield (4.30pm).
Red Devils legend Ince watched on from the sidelines on Wednesday night and felt Rashford was ill-prepared to play when he came on.
Speaking on BT Sport, Ince said: “He’s more or less started every game. I don’t think he’s used to being a substitute
“I don’t think he knows the process of being a substitute …how vigorously you warm up, do the right things. Do you think you’re going to get on the pitch? So it’s kind of a mindset where you think ‘well hopefully I’m not going to get on’.
“All of a sudden he’s thrust into the action because they need a goal.
“He is a talisman, he is the main man, and 16 minutes down he’s getting dragged off again with a bad back which hopefully he’ll be ok for the Liverpool game.”
Paul Scholes is confident Rashford will be available on Sunday, though, as he added: “I think it was actually something he picked up in the Manchester City game where he got a bit of a knock on his back.
“But he played the Norwich game, was excellent Saturday so this was an opportunity for him to have a break tonight.
“Ok it didn’t quite work out, he had to come on because United were struggling.
“I think he’ll be ok, he walked off alright and maybe a bit of ice, couple of days rest, he’ll be fine for Sunday.”