Getzlaff helped out on Nick Bonino's power-play goal with 4 1/2 minutes left in the second period, which broke a scoreless tie.
Anaheim was on a 5-minute power-play after Detroit's Justin Abelkader was given major penalty and a game misconduct for a hit to the head of Toni Lydman.
Getzlaff doubled the Ducks' advantage early in the final frame with a shorthanded goal, and Emerson Etem and Matt Beleskey both scored in the third period as the Ducks won going away.
Hiller made 23 saves as Anaheim grabbed the advantage in the Western Conference quarterfinal series.
Jimmy Howard turned aside 25 shots for the Red Wings, who were blanked in a playoff game for the first time since 2007.
Game 4 is set for Monday in Detroit.