Rinne has 29 shutouts in his five full seasons with the Predators, who snapped a three-game losing streak. He stopped 24 shots, just of six of them coming in the third period.
Nashville scored twice in each period with Patric Hornqvist opening the action on the power play 6:07 into the contest.
David Legwand also scored in the first and both Richard Clune and Zach Boychuk contributed goals in the second.
Wilson then scored on a backhander 4:43 into the third and delivered another power-play goal with 3:21 remaining. They were his sixth and seventh of the season.
Devan Dubnyk allowed two goals on 13 shots in the first period and was then removed in favor of Yann Danis.