Canada's Davis Cup defence ended at the quarter-final stage as Finland came from behind to win 2-1 and progress to the last four in Malaga, Spain.
But Otto Virtanen levelled for Finland by beating Gabriel Diallo 6-4 7-5.
He and Harri Heliovaara then defeated Alexis Galarneau and Vasek Pospisil in the deciding doubles.
The Finns won 7-5 6-3 to book a semi-final spot for the first time in the nation's history.
Canada had been dealt a blow before the start of the tie when top-ranked player Felix Auger-Aliassime was forced to sit out with a "lower body" issue, while Finnish number one Emil Ruusuvuori was also missing through injury.