Pavel Bure, known as the "Russian Rocket," scored twice for the Canucks, who took a 2-1 lead in the best-of-seven series with the next two games in Vancouver on Sunday and Tuesday.
Vancouver goalkeeper Kirk McLean stopped 29 shots for his third shutout of the playoffs.
"The win's the important thing," McLean said. "The shutout's the bonus."
Greg Adams scored in the second period and Martin Gelinas scored the last Vancouver goal in the game's final minute when Toronto was two men short.
"Kirk was the difference," Canuck Coach Pat Quinn said. "We weren't as good as I would have liked us to have been. We weren't sharp. We were lucky in a lot of cases tonight, and Kirk was really good."
The victory was the fourth in seven playoff games for the Canucks, who are 6-2 on the road after splitting two games at Toronto earlier this week.
Vancouver has won nine of its last 11 games in getting within two victories of its first Stanley Cup final since 1982.
"Vancouver is playing well, so we have to play well the rest of the games," Toronto's Wendel Clark said. "We're not playing our game the way we can, but don't take anything away from a club that's playing well."
The game deteriorated into an ugly brawl at 16:50 of the third period after a check by Vancouver wing Tim Hunter on Doug Gilmour of the Maple Leafs. Several fights broke out and the altercation continued after Gilmour went after Cliff Ronning, the smallest Vancouver player.
Referee Andy van Hellemond sent both teams to the dressing room after fans littered the ice with debris, including boxing gloves and a dead fish.
Emotions spilled over into the press box when George McPhee, vice-president of hockey operations for the Canucks, smashed a window, cutting a statistician.
When the teams returned 10 minutes later, four players were ejected, including Gilmour.
Quinn said the brawl was "calculated by the Maple Leafs. They pulled that last series against San Jose."
Toronto outshot the Canucks, 15-9, in the opening period, getting most of the scoring chances during three power plays. Vancouver killed off a two-man disadvantage for 1:29.
"It's a win and that's all that matters," Canuck defenseman Gerald Diduck said. "We didn't give them a lot of odd-man chances on the rush. That allows us to stand up at the blue line."
Bure opened the scoring with a spectacular effort at 13:25 of the first after a neutral-ice giveaway by defenseman Bob Rouse.
Bure scored on another breakaway in the third after Vancouver forward Murray Craven blocked a shot.