Nick Kyrgios watched proceedings from the sidelines.
With Pospisil defeating John Millman 7-6 (9-7) 6-4 in the opening match, Australia relied on a return from De Minaur to get himself out.
"Yeah, I understand it's the first time and you're going to have hiccups and you've got to learn from it". "He couldn't play. That's just the way the cards fell".
Britain was not playing Andy Murray, who on Wednesday came from behind to win his match against Tallon Griekspoor in three sets.
Shapovalov played superbly in the opening set to overpower fellow 20-year-old De Minaur but he could not sustain his level and the wily Australian, ranked 18th, hit back.
Thankfully there were a few hundred Canadian fans inside the sparsely populated 12,500-seat centre court at the Madrid venue that is hosting the first edition of the revamped Davis Cup.
Australia face Canada in the first of the quarter-finals from 17:00 GMT on Thursday.
"The sky's the limit".
"The format changed but we still came here to give absolutely everything we could, and we did that". "It's been a great run so far".
Organizers at the Davis Cup Finals are switching to earlier start times to minimize the chance of late finishes.
Canada forfeited their doubles rubber in Tuesday's Group F clash against the United States, also after winning both singles rubbers.
Roberto Bautista Agut earlier had comfortably beaten Nikola Mektic 6-1, 6-3.
Some top stars skipped matches to lessen their playing schedules, showing the need for reform, but tennis is still torn over whether Pique's "World Cup of tennis" will ruin or save the tournament.
Glory for Kevin Krawietz and Andreas Mies in the doubles capped off a 2-1 triumph for Germany, with Garin's ultimately the only sets won by Chile at the Finals.
Britain made a sturdy delivery after Kyle Edmund played brilliantly to gaze off Mikhail Kukushkin 6-3, 6-3 but Dan Evans became overwhelmed 5-7, 6-4, 6-1 by Alexander Bublik to leave the tie within the balance.
Serbia will play Russian Federation on Friday at an earlier time of 0930GMT while Spain's match against Argentina and Britain's against Germany will begin at 1630GMT after organisers brought forward both sessions by half an hour.
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