Anaheim will host what will be the first Game 7 in this year's NHL playoffs on Sunday.
The Red Wings appeared headed for a win in regulation before the Ducks scored twice in the final 3:28 of regulation.
Zetterberg, however, made up for that collapse by putting a slap shot past Jonas Hiller to preserve Detroit's season. It was his second goal of the contest.
Four of the six games played in the Western Conference quarterfinal series have been decided in overtime with each team winning twice.
The game was deadlocked at 1-1 heading into the third period, but Zetterberg gave the Red Wings the lead 6:19 into the stanza and Daniel Cleary boosted the Detroit advantage 5 minutes later.
Anaheim rallied on late goals from Emerson Etem and Bobby Ryan, whose game-tying score came with 2:37 left.
Jimmy Howard made 34 saves for the Red Wings. Hiller stopped 25-of-29 shots for the Ducks.