Shapovalov was a set away from booking Canada’s spot in the last four after Pospisil beat John Millman 7-6(7) 6-4, but Alex De Minaur recovered to beat the Canadian 3-6 6-3 7-5 and force a doubles decider.
Starting at 11pm local time, Shapovalov and Pospisil took down John Peers and Jordan Thompson 6-4 6-4 as Canada became the first nation to reach the semis.
Canada will face the winner of Russia vs Serbia for a place in Sunday’s final.
Opinion: Vaccine passports should be mandatory for players
Pospisil and Shapovalov have played in every rubber so far with captain Frank Dancevic short of options.
"It's been a pretty surreal week," said Dancevic. "I'm so proud of these guys. I mean, these two guys playing all week.
"Now that we're here, we're going to try to go all the way."
Australia, who won the last of their 28 Davis Cup crowns in 2003, were surprisingly without Nick Kyrgios, who won both his singles matches in the group-stage wins over Colombia and Belgium, and he was sorely missed.
Kyrgios, who banged down 22 aces against Belgian Steve Darcic on Wednesday, was sidelined with a collarbone injury.
Friday’s remaining three quarter-finals
Serbia's clash with Russia starts at 9:30am GMT on Friday, with the matches between Britain and Germany and hosts Spain and Argentina beginning at 16:30 GMT
Djokovic sends Serbia into Davis Cup quarter-finals
Novak Djokovic beat France's Benoit Paire to send Serbia into the Davis Cup quarter-finals.
The result means France are eliminated from the competition and Argentina progress as one of the best second-placed teams.
Djokovic coasted through the first set, breaking Paire early and holding on serve, before breaking early in the second and taking full control.
The Serbian world no 2 then breezed to a second set victory to take the match, barely breaking a sweat and looking very comfortable on court as he put his country into the next round.
Serbia's Filip Krajinovic had early given Serbia 1-0 lead by winning their opening singles rubber, beating Jo-Wilfried Tsonga 7-5 7-6.
Serbia and Argentina join Russia, Canada, Spain, Australia, Great Britain and Germany in the quarters.
With additional reporting from Reuters
French Open 2021 - How to watch Roland Garros and more key info
Federer in, Serena out - What does Madrid entry list mean for legends' seasons?