Zetterberg scored 1:49 into the game and later assisted on Valtterii Filppula's tally in middle frame, which put the Red Wings ahead 3-1.
In between Justin Abdelkader scored a shorthanded, unassisted goal and the seventh-seeded Wings went on to take the deciding Game 7 of their Western Conference quarterfinal series over the second-seeded Ducks.
Jimmy Howard stopped 31-of-33 shots for Detroit, which now goes on to face the top-seeded Chicago Blackhawks in Round 2.
Emerson Etam and Francois Beauchemin scored for Anaheim, which fell to 2-3 all-time in Game 7s.
Jonas Hiller turned aside 29-of-32 shots for the Ducks.
Duggar sisters unveil Christian dating rules in new book
Ohio bar shooting arrested, charged with murder