Miller scored his ninth goal of the season unassisted with 2:59 remaining in the extra session. It came moments after Montreal’s Josh Anderson failed to convert a breakaway attempt.
Nick Suzuki forced overtime with a power-play goal at 19:03 of the third, his sixth. He scored with Montreal on a two-man advantage as goaltender Jake Allen, who stopped 25 shots, was on the bench.
Adam Gaudette and Nils Hoglander scored in regulation for Vancouver (16-16-2). Goalie Thatcher Demko stopped 29 shots to register his eighth win in nine starts.
Corey Perry had the other goal for Montreal (13-8-9), which fell to 0-6 this season in overtime. The Canadiens were coming off 4-3 overtime loss to Winnipeg on Wednesday to cap a six-game road trip (2-2-2).
Montreal opened a six-game homestand with its first contest at the Bell Centre since a 7-1 victory over the Winnipeg Jets on March 6.
Montreal will host Vancouver again Saturday night. Prior to the contest, the Canadiens honoured long-time equipment manager Pierre Gervais, who recently appeared in his 3,000th NHL game.
Allen and Perry combined to stake Montreal to a 1-0 first-period lead.
Perry opened the scoring on the power play at 4:15. After Jeff Petry’s shot from the point hit the post, Perry knocked the rebound past Demko for his sixth of the season.
Allen, appearing in his 300th career NHL game, was solid throughout the frame as Vancouver outshot Montreal 11-8.
But Gaudette pulled Vancouver even at 2:25 of the second. He redirected Antoine Rousell’s centring attempt past Allen for his fourth of the year,
Hoglander gave Vancouver its first lead of the game, deflecting the puck past Allen at 6:37, his sixth of the season.
Vancouver was minus forward Tanner Pearson (lower-body injury). He was hurt in the third period Wednesday against Ottawa and is expected to be out at least four weeks.
Pearson has 11 points (six goals, five assists) in 33 games this season. Forward Jimmy Vesey made his debut with Vancouver after being claimed off waivers from Toronto on Wednesday.