Joe Colborne scored for Calgary in the first round of the tiebreaker, and Yannik Weber evened it in the second for the Canucks.
The shootout remained knotted until the fifth when Higgins switched from his backhand to his forehand side and slid a shot between the pads of Karri Ramo for the game-winner, handing the Canucks just their second victory in 10 games.
The game was just 2 seconds old when a melee broke out between the teams, with five fights ongoing simultaneously. There were two more altercations before the period was finished -- the two squads combined for 204 penalty minutes for the game.
Television cameras showed Vancouver Coach John Tortorella trying to force his way into the Flames' locker room between periods, with a scrum breaking out in the narrow hallway outside the dressing room.
Ramo finished with 31 saves.