Montreal won for just the third time in eight games and handed the Coyotes their second straight setback.
The game was deadlocked heading into the third period but Bergeron put Montreal in front for good with a slapshot that bested Ilya Bryzgalov 5:10 into the stanza.
Glen Metropolit added a power-play goal for the Canadiens 62 seconds later but Phoenix gave itself a chance midway through the third when Paul Bissonnette tipped in a shot by Shaun Heshka.
The Canadiens protected their one-goal lead until Brian Gionta scored into an empty net with 41 seconds to play.
Carey Price made 18 saves for Montreal, which defeated Phoenix for the sixth straight time.