Price has 290 wins, 24 behind Jacques Plante for most in Montreal history. The shutout was Price's first of the season and the 41st of his NHL career.
"I had my cake and I ate it too. So it's a good day," Price said about the triple success of passing Roy, getting a shutout and defeating the rival Bruins.
WATCH: [All Canadiens vs. Bruins highlights]
Brendan Gallagher and Max Domi scored 1:21 apart in the first period for the Canadiens (6-2-2), who are 2-1-1 in their past four games.
"I think we had to bounce back from what we felt was a tough loss [4-3 at the Buffalo Sabres on Thursday], one that got away from us," Montreal coach Claude Julien said. "It was important to come in here and try and gain two points. First period, the first five, six minutes, we were on our heels, and then we got playing a little bit and got a lead.
"… We did what we had to do in the third period to protect the lead and try to win this game."