Koivu put a wrist shot past Jason LaBarbera 2:43 into the extra period to bring the Wild their seventh straight victory.
Minnesota's winning goal came moments after a penalty to the Coyotes' Antoine Vermette expired. Zach Parise sent a pass to Koivu, who sped to the net from the right circle and beat LaBarbera for his ninth goal of the season.
Niklas Backstrom picked up the win by making 24 saves and Minnesota moved into a tie with Vancouver for first place in the Northwest Division.
Suter tied it with 55 seconds left in the third period after Phoenix had rallied from a two-goal deficit to take the lead.
Michael Stone gave the Coyotes a 3-2 advantage 1:52 into the third period. LaBarbera stopped 33 shots for Phoenix, which has lost seven straight games.