Dallas (20-14-7) outshot the Red Wings 45-27, but went 0 for 7 on the power play.
The Stars had the first 12 shots on goal, but Zetterberg scored on the Red Wings' first shot 7:04 into the game.
Dallas tied it on Rich Peverley's short-handed goal with 7:13 left in the period.
Detroit's Justin Abdelkader missed a penalty shot with 5:58 left, but Andersson put the Red Wings back ahead for good lead with a shot from the right circle over starting goalie Dan Ellis with 1:55 to go in the first.
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