Trotz is the only coach Nashville has ever had. He has led the Predators to six playoff appearances since the team began competing in the 1998-99 season.
Nashville is headed for another post-season trip, having already clinched a playoff spot. The Predators won for the third time in four games and handed the slumping Red Wings their eighth loss in 10 outings.
Nashville moved a point in front of Detroit for the fourth seed in the playoffs, which brings with it home-ice advantage in the opening round.
Detroit's only goal came with 3:03 to play in regulation from Jiri Hudler.
Jimmy Howard returned to the Detroit goal for the first time since March 19 after suffering from a groin injury. He made 23 saves.