Miller scored his ninth of the season moments after Montreal's Josh Anderson failed to convert a breakaway attempt.
Miller drove down the left side, cut across the front of the net and put a backhand shot past goalie Jake Allen to give Vancouver its seventh win in eight contests.
''It was not really textbook,'' Miller said. ''I was on the ice for a really long time but I wasn't really doing a whole lot . . . I still felt fresh.
''It's one of those things I knew I was going against a forward, then when I got by the first guy it's a heat-of-the-moment move that doesn't happen very often. I was pretty lucky to get it by and obviously it was a big goal for us but that's not going to happen too often.''
Nick Suzuki forced overtime with a power-play goal with 57 seconds left in overtime with Montreal on a two-man advantage with Allen off for an extra attacker.
''It wasn't like a deflating goal,'' Miller said. ''They had a six-on-four with some of the best shooters in the league up top.''
Adam Gaudette and Nils Hoglander scored in regulation for Vancouver, and Thatcher Demko stopped 29 shots for his eighth victory in nine starts. The Canucks are 16-16-2.
''It was a good team effort right from the drop of the puck,'' Canucks coach Travis Green said. ''The team checked well, we just played a good road game. That's the type of hockey we need to play to have success and we've been talking about it for a long time now. It's nice to get the win.''
Corey Perry also scored for Montreal, and Allen made 25 saves in his 300th NHL game. The Canadiens are 0-6 in overtime this season.
''Obviously at one point it's mental but what we need is to bury one and get that over with,'' Montreal interim coach Dominique Ducharme said. ''We do that tonight, probably everyone's talking about what a character team we are coming back and tying the game.
''Our guys, for sure they think about it when the time comes. I'm sure a guy like Andy going on a breakaway, he's probably putting more pressure on himself thinking, `I'm going to end it this time.' Yeah, that's apart of it but we have to stick with it and next time we'll get it.''
The teams will complete the two-game series Saturday night.
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