Price turned aside 11-of-12 shots in the third period as the defending Stanley Cup champions battled to erase a two-goal deficit.
Boston's Tyler Seguin dented Price with a goal with 55 seconds remaining, pulling the Bruins to within 3-2. But Tomas Plekanec sealed the victory with an empty-netter as the Canadiens followed up on Thursday's 2-1 win in Boston.
Brian Gionta, Lars Eller and David Deshamais also found the net for the Canadiens.
Milan Lucic scored late in the second period for the Bruins, who fell to 3-7-0 to start the season.
Tuukka Rask made 23 saves for Boston.