Brown slid the puck over to Paul and he beat goaltender Jussi Olkinuora 6:26 into the extra frame.
Maxime Comtois and Adam Henrique also scored for Canada, which looked down-and-out after losing its first three games in Riga before going on an improbable run to the top of the podium.
Brown assisted on all three of Canada’s goals in the final to end the championship with a tournament-high 16 points (two goals, 14 assists).
Mikael Ruohomaa and Petteri Lindbohm scored for Finland.
Andrew Mangiapane, whose arrival midway through the preliminary round provided a spark for Canada, was named player of the tournament. The Calgary Flames forward had seven goals — including the winners in Canada’s quarterfinal and semifinal victories — and four assists in seven games.
Darcy Kuemper made 29 saves.
Earlier, the United States beat Germany 6-1 in the claimed bronze with a 6-1 win over Germany. Cal Petersen made 33 saves for the Americans.