Rashford was called in at the last minute to start against Midtjylland in the Europa League on Thursday, finding the net twice as the Red Devils soared to a 5-1 win over the Danish champions.
The young striker was rewarded for his fine display in midweek with his first Premier League start against Arsenal and captured the imagination again by netting another double to dent the Gunners' title hopes.
Rashford, who was one of three debutants against Arsenal alongside substitutes Timothy Fosu-Mensah and James Weir, has since received widespread acclaim for his efforts but Carrick is certain he won't let it all go to his head.
"I have seen some debuts but none quite like that," Carrick told MUTV. "It is an unbelievable start for him [Rashford].
"He has just got to keep his head down and keep going. There is going to be a lot of attention on him now, people are going to be looking at him because he has thrown himself straight into the spotlight – which is great to see.
"He is a level-headed guy and I am sure he will deal with it well.
READ MORE | Rashford: Arsenal brace was 'amazing'
"For all of us, just to see what it means to him and the other new boys who are new to the team, after working hard for so long and for so many years, is great and everybody responds to that.
"Obviously Marcus is going to get the headlines but I though [Guillermo Varela] was terrific at right-back and especially after getting a yellow card against somebody like [Alexis] Sanchez.
"He was my wing-man and my support just outside of me.
"I thought Timmy [Fosu-Mensah] did well when he came on - to come on in a game like that at left-back which is not really his position.
"There are so many positives to take from it. I think there was a real togetherness in the squad and I think the fans responded to that.
"Seeing somebody like Marcus [Rashford] doing what he has done this week has given everybody a huge boost, so it is great to see everybody so happy."