"They have a lot of confidence in their goaltender and they play like it," Canadiens left wing Michael Cammalleri told reporters.
The shutout was the 17th in 299 career games for Miller, who has been selected to play for the U.S. Olympic team in Vancouver.
Tim Connolly scored the game's only goal during the second period on a tip-in to give the Northeast Division-leading Sabres their fourth consecutive win.
Canadiens netminder Carey Price stopped Connolly's first attempt, but the Buffalo center deflected the puck past Price to score 6:56 into the period.
"That's how you score goals in this league and that's one thing they did a little bit better," said Price, who had 29 saves. "It came right back on his stick and he was still there."
The loss was especially disappointing to the Canadiens, who were coming off a 6-1 run in seven consecutive road games.
"When you're successful on the road, you want to come back home and play a simple game." said Montreal's Brian Gionta.
"You don't want to be too fancy and I think we were too fancy tonight.
"We were trying to pull up and make plays but a team like Buffalo has good back pressure and they work hard and get back and they broke up some of our odd man rushes."
(Reporting by Gene Cherry in Salvo, North Carolina; Editing by Greg Stutchbury)