Nilstorp, a 29-year-old veteran of the Swedish Elite League, notched his first NHL win in his fourth North American start as Dallas snapped a two-game losing streak.
Antti Niemi coughed up three goals on 28 shots for the Sharks, who completed a 1-4-1 road trip.
Patrick Marleau got San Jose's only score.
Ray Liotta sues skin care company over use of likeness
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