Weber scored 80 seconds after Wilson as the Predators jumped to a 2-0 lead just 3:23 into the game. Weber added a second goal and Wilson picked up an assist.
Pekka Rinne turned aside 32 shots for the winners.
San Jose goalie Antti Niemi didn't last through the first period. He was pulled after yielding three goals on seven shots.
Thomas Greiss relieved for the Sharks and turned aside 11-of-13 shots the rest of the way.
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