The Twins announced the move before Wednesday's game.
Van Mil is a right-hander from the Netherlands who turns 26 later this month. He was designated for assignment to make room for Fuentes when the deal was made last Friday.
Van Mil had a 6.37 ERA in 23 relief appearances for Double-A New Britain this season. Injuries have hindered his progress since he was signed by the Twins in 2006.