Denis Shapovalov got things started for the Canadians, who made their Davis Cup debut in 1913, by beating Thanasi Kokkinakis 6-2 6-4.
The 23-year-old - 18th in the ATP rankings - pulled ahead early against his opponent, breaking the Australian twice to take a 4-0 lead.
It took just 32 minutes to settle the opening set.
Shapovalov missed his first opportunity to serve for the match at 5-2 in the second, instead allowing Kokkinakis to break back and eventually cut his lead to 5-4.
He made no mistake the second time of asking, however, and set up Felix Auger Aliassime for the winning match.
The Montreal native, ranked sixth, had beaten opponent Alex de Minaur in straight sets earlier this year in Cincinnati and hoped to recreate his success to snap Canada's 109-year streak without a trophy.
He did so handily, hitting six aces and 16 winners without seeing his serve broken en route to a 6-3 6-4 victory to clinch the title for a Canadian team that also included Vasek Pospisil, Alexis Galarneau and Gabriel Diallo.
"The emotions are tough to describe," he told the Davis Cup media channel. "I think all of us here, we've dreamt of this moment.
"I mean these guys around me, except Vasek, he's a little bit older than me, but we grew up together form the ages of seven, eight years old, back in Canada dreaming of being on this stage, winning these types of matches, winning the Davis Cup, so it's really a dream come true.
"For me personally and all the team, that's what we play for, that's what sport is for, tennis is for, so it was a great moment. A great moment for myself and the country."