Dogs, not cats, are the solution. My 2 Boxers don't allow any mouse or rat come near my garage. They chase and kill them mercyless, but never eat them.
Just bring the corpses to show us their good work.
My cats, well feed (as the dogs are, by the way), absolutely don't care about the rodent creatures. To be sincere, I guess they're somewhat afraid of them.
Fabio.