Marrow became manager Graham Kavanagh's eighth summer signing on Tuesday following his recent release by Blackburn and is confident the Cumbrians, relegated in May, can bounce straight back to League One.
The 24-year-old told Carlisle's official website: "As soon as I heard what Kav had to say I really wanted to be a part of it all.
"I really liked what he had to say. He's trying to build a successful squad to go straight back up and I think we've got a great chance of doing that next season.
"You don't want to come to a club that might go down, or isn't showing any ambition, but from what the manager has said it's a case of nothing apart from going back up is acceptable.
"Everybody is pushing for promotion and after seeing the lads, and the way they are, I think there's every chance of that."Marrow, a product of Blackburn's academy, had loan spells at Oldham and Crystal Palace, before making a permanent move to the Eagles.
He had further loan spells at Preston and Fleetwood before rejoining Blackburn last summer, but following another temporary stint at Fleetwood at the beginning of this year failed to win a new deal at Ewood Park.