Roared on by thousands of Finnish fans, Otto Virtanen and Harri Heliovaara beat Vasek Pospisil and Alexis Galarneau 7-5, 6-3 to move in to the semi-finals for the first time.
Milos Raonic, playing in the event for the first time in five years, had given Canada the perfect start as he dispatched lowly-ranked Finn Patrick Kaukovalta 6-3, 7-5 thanks to 18 aces.
Both teams were hit by injuries with Canada's top-ranked player Felix Auger-Aliassime unable to take to the court while Finland's number one Emil Ruusuvuori was also ruled out.
But 171st-ranked Virtanen stepped up for his country with an impressive defeat of fellow 22-year-old Gabriel Diallo to take the tie into a deciding doubles rubber.
Virtanen again produced some stunning tennis while Canada's hopes were not helped by an injury to Pospisil who battled on.
A service break in the eighth game of the second set proved enough and Finland sealed the win to set up a clash later this week against the Czech Republic or last year's runners-up Australia.