Rashford, 18, has shone since being promoted from the club's academy this season, scoring six goals in his 11 appearances, including a stunning strike in Wednesday's FA Cup quarter-final win over West Ham United.
“Not in his wildest dreams could he have predicted the last two or three months that he's had,” Carrick told the club's website (www.manutd.com).
“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 (against West Ham) - we'll welcome more of them.”
United, who are fifth in the Premier League, are vying with local rivals Manchester City for a spot in next season's Champions League.
City, who are fourth and hold the final qualifying spot, visit Chelsea on Saturday, while United are home to bottom club Aston Villa, whose relegation could be confirmed this weekend.
Original source: Rashford keeping his feet on the ground