Zetterberg's goals all came on power plays as he reached 30 goals for the second straight season and helped the Red Wings sweep the four-game season series with the Coyotes.
Zetterberg also had an assist on Jason Williams' empty-net goal in the final minute.
Dominik Hasek stopped 20 shots for Detroit.
Mike Zigomanis got the Coyotes on the board 25 seconds into the third.
Mikael Tellqvist made 23 saves for Phoenix.