Hiller, who in his second NHL season has replaced Jean-Sebastien Giguere as the Ducks No. 1 goaltender, made 35 saves and produced the first shutout of this year's post-season. The veteran Giguere was the MVP of the 2003 playoffs.
San Jose, having been eliminated in the second round each of the last three seasons, got off to a rocky start after earning the No. 1 seed in the Western Conference.
Anaheim became one of just two road teams this year to win the first game of an opening-round series. The New York Rangers won on the road Wednesday against Washington.
San Jose wasted four power-play chances during the scoreless second period, three of those opportunities coming when Scott Niedermayer was sent to the penalty box.
Niedermayer then scored the first goal of the series, a power-play tally that came 5:18 into the third period.
Ryan Getzlaf gave the Ducks some insurance with 2:25 to play by putting a wrist shot past Evgeni Nabokov.
Hiller's performance was made even more impressive by the fact that he gave up four goals in 29 minutes and was removed from the Ducks' final game of the regular season -- a 5-4 shootout loss to Phoenix.
Thursday's game was the first NHL playoff game between two California teams since the Los Angeles Kings won in seven games over the Oakland Seals in the opening round in 1969.
Notable deaths of 2014 [PHOTOS]