The Sharks were playing for only the second time this season, having won their other contest against Phoenix six days ago. The loss by San Jose leaves Philadelphia, Toronto, Washington and Detroit as the only unbeaten teams in the NHL.
Hiller, who has been in goal for all three of Anaheim's games this season and has won two of them, had a career-best five shutouts last year.
The lone goal was scored by Maxime Macenauer on a tip-in of a shot by George Parros with 6:01 to go in the first period.
Thomas Greiss, filling in as goaltender while Antti Niemi recovers from a knee injury, took the loss while making 22 saves.