Andrew Cogliano scored in regulation for Anaheim, which has won five in a row and gone nine games without a regulation defeat (7-0-2).
Bobby Ryan also scored in the shootout, while Hiller denied both Dallas shooters he faced.
Ray Whitney, in his second game back after missing 16 due to a broken foot, scored the only goal for the Stars.
Kari Lehtonen, making his sixth straight start, had 29 saves.
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