Michael Cammalleri and Brian Gionta also scored, and Tomas Plekanec had three assists for Montreal. The Canadiens snapped a two-game losing streak and won for just the third time in eight games.
Paul Bissonnette scored his first career goal, and Robert Lang also scored for Phoenix. The Coyotes have lost four of five.
Bergeron made it 2-1 at 5:10 of the third after Plekanec won a faceoff to the left of goalie Ilya Bryzgalov. Bergeron circled back to the blue line before striding to the top of the left circle and firing a wrist shot just under the crossbar
Metropolit made it 3-1 at 6:12, standing untouched on the right doorstep and taking a pass from Plekanec before flipping the puck into the net.
Bissonnette pulled the Coyotes within a goal at 9:30, redirecting Shaun Heshka's long shot past Price. Replays indicated Bissonnette had directed the puck into the net with his skate, but video officials upheld the score after a lengthy review.
Gionta added an empty-netter with 41.8 seconds to go.
Cammalleri gave Montreal a 1-0 lead at 7:07 of the first period, taking a pass from Plekanec just to the right of Bryzgalov and scoring before defenseman James Vandermeer was able to get in position.
Lang tied it at 10:54 of the period after Price was knocked backward by the stick of teammate Paul Mara. Lang took a pass from Radim Vrbata from behind the net and shot into the open top of the net as Price lay prone on the ice.
Notes
The announced attendance was 10,064, the fourth-highest total in nine home games this season. … Heshka's assist was his first NHL point. … The Coyotes have lost two straight for the fourth time but have yet to lose three in a row. … Montreal has not lost in Phoenix since Dec. 9, 1998. The Canadiens are 10-2-4 against the Coyotes since the franchise moved to Phoenix before the 1996-97 season. … Phoenix defensemen Ed Jovanovski and Zbynek Michalek missed the game because of lower-body injuries.