Bernier was at his best in the second period as the Coyotes outshot Toronto 15-6, including a sliding stick save on a point-blank shot by Oliver Ekman-Larsson.
Jake Gardiner, Phil Kessel and Nazem Kadri each provided a goal and assist for the Leafs, who posted a fifth consecutive win following four straight losses for the first time since 1967.
Carl Gunnarsson also scored for Toronto.
Mike Ribeiro and Antonie Vermette each recorded a goal for Phoenix, which had its two-game winning streak snapped.
Mike Smith made 25 saves.