It was the first victory for the Flames in three tries this year and it came in the franchise-record 263rd career win for goaltender Miikka Kiprusoff. He allowed a goal to Andrei Kostitsyn 1:47 into the contest and then shut the door the rest of the way.
Kiprusoff, who is in his 11th NHL season and eighth with the Flames, made 35 saves.
Bourque tied it at 7:15 into the first on the power play and Niklas Hagman put Calgary in front for good with a power-play goal that came with 8:22 to go in the opening stanza.
David Moss scored in the first period and Bourque put a wrist shot past Carey Price in the second. Price stopped 17 shots in what was Montreal's home opener.
The Canadiens split their first two games on the road.