Rashford opened the scoring at Upton Park with a curling effort that left keeper Darren Randolph helpless in the West Ham net.
Marouane Fellaini eventually scored a second as Man United booked their semi-final spot with a 2-1 win. Carrick was full of praise for Rashford following the clash.
“It’s incredible. It’s so good to see young lads come in, do well and take their chance. Not in his wildest dreams could he have predicted the last two or three months that he’s had,” Carrick said.
“He’s still got his feet on his ground though. He keeps coming in, he doesn’t change, he works hard, he listens, he’s willing to learn and I think he’s got a great chance of staying at the top for a long time.
“I don’t think it’s a flash in the plan, in and out. I think he’s here to stay and goals like that… we’ll welcome more of them!
“We haven’t done well enough in this competition over recent years and we’re well aware of that. We’re desperate to go all the way this time.”
Promising talents have come and gone at Old Trafford – Federico Macheda immediately springs to mind – but there is a sense that there is something different with Rashford.
His goal was spectacular but his all-round game was equally as impressive and he played with a maturity that belied his years. It will be exciting to see how he develops at United in the coming years.