Making his NHL playoff debut after being called up Wednesday from Providence of the American Hockey League, the 23-year-old Fraser jumped into a scramble in front of goalie Carey Price and slid the puck under him into the net.
"It's something I dreamed about many times on the outdoor rinks growing up," Fraser said. "It's every kid's dream to score in overtime."
"It was bouncing around and I was just swatting at it as hard as I could and hoping it would hit something," Fraser said. "Price is such a good goalie. He doesn't give out a lot of rebounds. I might have got lucky there, but the puck wound up in the back of the net."
Game 5 is Saturday night in Boston.
"We knew how tight this series would be," Boston coach Claude Julien said. "It doesn't mean any more than that we've tied the series. It's been tight every game.
"It was important to get back in the series, but I don't think we've played our best hockey. We've played hard, but I've seen us play better. You hope the win here will help us get better and we'll go from there."
Tuukka Rask made 33 saves for Boston, and Price stopped 34 shots.
"This was a hard-nosed game between two well-prepared teams," Canadiens coach Michel Therrien said. "We lost a battle in front of the net on the winning goal, but I'm pleased with our team's effort."
The teams played their first scoreless period in regulation time in the series in a tight-checking first marked mainly by icings and missed passes.
Boston's Carl Soderberg had the best chance in the final minute after Montreal defenseman P.K. Subban's blind back pass was intercepted, but Price got a piece of it as it went off the crossbar.
Another goal-less period followed as shooters misfired, particularly Montreal's Max Pacioretty on a setup from Thomas Vanek, and both goalies held their ground. Each team went on the power play once, but even that failed to break the deadlock.
The Canadiens went on attack in the third, peppering the Boston net and outshooting the Bruins 14-7. But the Bruins got a power play and Reilly Smith hit a goal post, only to see Brian Gionta stoned by Rask alone in front of the net at the other end a moment later.