It was the second multigoal game of his career (Dec. 10, 2016).
"I got a couple of bounces," Shore said. "It helps, and it's human nature to start feeling good about yourself, getting the breaks."
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Joe Pavelski, Evander Kane and Timo Meier scored, and Martin Jones made 18 saves for San Jose (8-5-3), which had won two straight.
"If you score three, you should be getting points," Sharks coach Peter DeBoer said. "We shot ourselves in the foot. We made some soft plays with the puck or some poor decisions."
Joe Thornton played in his 1,500th NHL game, becoming the 19th player to reach that milestone. He had an assist to give him 1,032 for his career, one behind Mario Lemieux for 11th place in League history.
Tyler Seguin had a goal and an assist, and Anton Khudobin made 33 saves for Dallas (9-6-1), which was coming off a six-game road trip (3-2-1).
"I think we've toughened up on the road trip," Stars coach Jim Montgomery said. "I think we've become resilient."