Marrow, 24, a free agent after leaving Blackburn, has signed a one-year deal with the Cumbrians.
Kavanagh told the club's official website: "We've had Alex in the building for about 14 days now and it's been on and off for one reason or another.
"You do worry, with the quality of player he is, that other clubs will step in when you haven't got the deal finalised.
"I think we did well because we managed to get him up here and he's been able to see the way we work and the styles we want to try and play.
"He felt wanted by us and when you have that as a player it goes a long way to helping you make a decision.
"I think he's one who will grow stronger and fitter and he'll be a big asset for us."
Kavanagh added that another of his summer signings, Gary Dicker, will be out for approximately two weeks after sustaining a thigh injury.