After the two teams battled through a scoreless regulation and overtime, Wilson finally broke the drought when he tipped a Ryan Ellis pass behind the Sharks' Antti Niemi for the decisive score.
Rinne turned aside all 25 shots he saw, including three in the overtime, en route to his second shutout of the season and 27th of his career as the Predators broke a two-game skid.
The Sharks, meanwhile, stumbled to their sixth straight loss.
Niemi stopped 24-of-25 Nashville stops in the setback.
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