The Predators faced a one-goal deficit heading into the final period, but Sergei Kostitsyn tied it and Wilson delivered the game-winning score just 44 seconds later.
Anders Lindback, who has been getting the playing time for Nashville since Pekka Rinne suffered a knee injury this month, made 25 saves. Lindback is 5-0-1 since Rinne went out.
Niclas Wallin and Dany Heatley scored for San Jose and Antti Niemi stopped 32 shots.
Nashville has won four in a row, its longest winning streak of the season.