The game matched two of the hottest teams in the league. Phoenix had won its previous five games and the Kings captured their sixth win in seven outings.
Both sides scored in the second, fifth and seventh rounds of the shootout. After Williams produced a goal to start the eighth round, Scottie Upshall was stopped by Jonathan Quick to bring the Kings the victory.
Los Angeles took a 2-0 lead in the first period, but the Coyotes eventually tied it on a power-play goal by Radim Vrbata 3:12 into the third period.
Quick made 29 saves in regulation and overtime. Ilya Bryzgalov stopped 21 shots for Phoenix.
The Kings climbed within a point of San Jose atop the Pacific Division.