Brendan Perlini added his first career goal, Martin Hanzal also scored and Mike Smith made 42 saves for the Coyotes, who were 0-3-3 since their last win against at Edmonton on Nov. 27.
Arizona scored four goals for the first time since Nov. 8 after totaling four in its previous four games.
Roman Josi scored and Pekka Rinne made 30 saves for the Predators, who have lost four of five (1-3-1).
The Coyotes took a 1-0 lead with 7:10 left in the first as Goligoski fired a one-timer from the top of the right circle for his first of the season.
Dauphin made it 2-0 less than two minutes later when he tapped in a rebound from the left side of the crease after.
