The Predators built a 3-0 lead through one period, but could not hang on behind the struggling goaltending of Carter Hutton.
Phoenix won its third in a row and picked up its first overtime victory of the year in three tries. Nashville has dropped back-to-back games.
None of the first eight attempts of the shootout produced a goal, but Boedker finally beat Hutton and Greiss then stopped Legwand to clinch the victory.
Hutton allowed the four goals in regulation on 25 shots. In his previous outing, a loss to St. Louis, Hutton gave up five goals on 26 shots.
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