Bertuzzi's second goal in the game, and sixth this season, broke a 2-2 tie 67 seconds into the 5-minute stanza.
Earlier, his fifth goal at 9:36 of the third period created a 2-2 tie.
Net-minder Jimmy Howard yielded two goals on 27 shots for the Red Wings, who have won two of three and have split their last four outings.
They played without forward Dan Cleary, who is expected to miss about a month with a shoulder injury.
Anaheim scored on second-period goals by Saku Koivu and Dan Sexton.
Jean-Sebastien Giguere made 28 saves on 31 shots for the Ducks, who have lost six of their last seven games.
Celebrity Couples of 2014 [PHOTOS]