The Predators ended a five-game losing streak, getting an outstanding effort from Rinne and two late goals after he had allowed the Wild to climb within one.
Cody Franson scored 4:23 into the contest and Patric Hornqvist tipped in a shot midway through the second period to give Nashville a 2-0 lead.
It looked as if that would be the final score but Chuck Kobasew deflected a shot from Martin Havlat past Rinne with just 1:54 to play.
Minnesota then pulled goaltender Niklas Backstrom in hopes of scoring the tying goal, only to have Joel Ward and Sergei Kostitsyn both put shots into an empty net.
Minnesota lost for the third time in four games. Backstrom stopped 30 shots to take the loss.
Celebrity Families of 2014 [PHOTOS]