With Pospisil defeating John Millman 7-6 (9-7) 6-4 in the opening match, Australia relied on a comeback from De Minaur to haul itself level.
Next Gen ATP Finals runner up De Minaur – Australia's star player in the tie with Nick Kyrgios sitting out – did it the hard way, coming from a set down to beat Shapovalov 3-6 6-3 7-5 and tee up a deciding doubles encounter.
But Canada was too good in the doubles – Pospisil picking up where he left off while Shapovalov bounced back from his loss in style via a 6-4 6-4 win.
A break of serve in the opening game against Australian counterparts Jordan Thompson and John Peers paved the way for the pair to claim the first set with ease.
Thompson and Peers looked set to take the match all the way when they earned a 3-0 lead in set two, only for Shapovalov and Pospisil to restore parity before pushing on to seal progression with their first match point.
Serbia plays Russia, Argentina faces Spain and Great Britain goes up against Germany in the other quarter-final ties.