Montreal scored the first three goals and controlled the play throughout, outshooting Boston 38-21.
Boston goalie Tuukka Rask stopped 37 shots and kept the Bruins from being routed.
The Bruins are off to their first 0-2 start in 10 years. Boston played again without captain Zdeno Chara, who sat with an undisclosed upper-body injury. The Bruins clearly missed their supersized defenceman.
Desharnais scored a power-play goal 1:25 into the game and Montreal scored again when Eller scored on a one-timer 2:15 into the second.
The Bruins felt they pulled within 2-1 with 13:20 left in the second when Loui Eriksson beat Price on a high backhand with 13:20 left in the second, but the goal was immediately waved off by one of the referees, who said Bergeron made contact with the Montreal goalie at the edge of the crease.
Boston coach Claude Julien challenged, but the call stood after a video review and disgruntled Bruins fans let loose with a loud chorus of boos. The Canadiens quickly quieted the crowd 1:07 later when Eller scored on another one-timer to put Montreal up 3-0.
Beleskey cut the lead to 3-1 with 5:34 left in the second on a shot that deflected off a defender and bounced between Price's pads.
Notes: Boston has allowed three straight goals three times already this season. ... Although the two clubs play four more times this season, Saturday was the only meeting in Boston. The Bruins host the Canadiens again Jan. 1 in the Winter Classic at Gillette Stadium in Foxborough. ... Boston C Ryan Spooner received a boarding major and game misconduct for a hit on Semin 3:30 into the third period.