Shane Doan scored twice in the third period for the Coyotes, who were shut out in their three previous games.
Bernier extended Phoenix's scoreless drought to 245 minutes, 32 seconds since rookie Rob Klinkhammer's goal against Jonathan Quick in the Coyotes' 5-2 win over the Kings on March 12. The Coyotes had been stopped on 133 consecutive shots before Doan's first goal.
Dustin Penner and Jarret Stoll also scored for the Kings, who have won 12 of 16 games and 11 of 12 at home. Stoll connected on a power play with 6:14 remaining while Derek Morris was serving an interference penalty.
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