Evgeny Svechnikov and Alexander Barabanov scored for the Sharks (17-29-11), who have lost three straight. Kaapo Kahkonen made 27 saves.
Skinner put the Sabres ahead 3-2 at 9:52 of the third. After winning a battle with Mario Ferraro behind the net, Tage Thompson poked the loose puck at the edge of the crease to Skinner, who shot into an open net.
Skinner's goal came after Kahkonen stopped Peyton Krebs on a penalty shot at 1:23 of the third.
"It really seemed like we got very demoralized after they went up 3-2. Whereas when it was 1-0 and 2-1 we kept playing," San Jose coach David Quinn said. "I thought we really played well the first two periods and controlled a lot of the play. I didn't love our third. They turned it up a notch. I thought we took a step back, and just miscommunication on the [third] goal. It looked like we got easily demoralized after they went up 3-2. We didn't do much after that."
Tuch scored into an empty net at 18:59 for the 4-2 final.
"I liked the way we committed to our game plan for the last period," Thompson said. "It wasn't pretty the first two, but we got ourselves to a spot to win a game, and I think the response after the second period was good. We committed to what we needed to do out there, and that's why we had success there in the third."