The victory capped a memorable comeback, even for a nation with as rich a history in hockey as Canada. Coach Gerard Gallant's team dropped its first three games of the tournament before it recovered en route to the title.
Brown slid the puck over to Paul and he beat goaltender Jussi Olkinuora 6:26 into the extra frame.
"We were getting better as the games went on and we knew we had the right guys in the room," Paul said in his postgame rinkside interview on TSN. "We just didn't have the bounces in the beginning of the tournament. Once those bounces started going our way, it put some wind in our sails and we just kept building every single game."
Maxime Comtois and Adam Henrique -- the team captain who plays for the Anaheim Ducks -- also scored for Canada, which looked down-and-out in Riga during the tournament's first week. But all along, Gallant believed in his team, which perhaps lacked the skill level of past Canadian rosters. Between several NHL stars declining invites due to the COVID-19 pandemic, and others in the midst of the Stanley Cup playoffs, Gallant worked with a collection of rising NHL prospects and veterans who did not qualify for the postseason.
All that said, after his team fell to 0-3, Gallant vowed he'd stick with the plan and told reporters that "we've played good hockey so far. We just haven't gotten the results."
The results came, and Canada never looked back. Brown assisted on all three of the team's goals in the final to end the championship with a tournament-high 16 points (two goals, 14 assists).
"Our only mindset in the overtime was that we came back during this whole tournament," Comtois said. "Nothing was easy. We came back from being down 0-3. We came back against Russia. We just told ourselves that we were going to do it again. We came out feeling loose, and we dictated the overtime."
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. His arrival was late because Calgary, like other Canadian teams, had its NHL season extended to make up several postponed games with the Vancouver Canucks, a franchise that was plagued by a lengthy COVID-19 pause at midseason.
The day began with Team USA defeating Germany 6-1 in the bronze-medal game. Cal Petersen, a goaltender for the Los Angeles Kings, made 33 saves for the Americans, who lost to Canada in Saturday's semifinals.
The biggest winner of the tournament -- in the long run -- could well be Gallant, a free-agent coach who led the Vegas Golden Knights to the Stanley Cup Final in 2018.
Gallant is a prominent name in the NHL's offseason coaching carousel, and interviewed with the New York Rangers before flying to Riga. This comeback in international competition -- especially with a young roster -- should make him even more marketable upon his return.
Even Henrique, who was waived by the Ducks this season only to be brought back to Anaheim when no other NHL team claimed him, could well use this as a springboard for the rest of his career.
"It's unbelievable, special to lead this group all the way to gold," he said on TSN. "I'm sure a lot of people counted us out early on in the tournament, but we kept believing in ourselves and in that locker room. We stuck with it right to the end and made it interesting all the way.
"It was a heck of a tournament."
The Associated Press contributed to this report.