For the second match in a row in Toronto, 18th-seeded Popyrin had to claw his way back after losing the first set, beating the talented Dane 4-6 6-2 6-3 on Saturday (local time).
It was just the second career meeting between the pair and squared the ledger at one apiece, Rune having won their only previous encounter in a tough three-setter on clay at the 2023 Rome Masters.
Popyrin, the defending champion in Canada, is looking to continue his impressive run at this tournament, his victory over Rune his ninth consecutive win at the event.
The Australian also recovered from the loss of the first set of his third-round match to score a 5-7 6-4 6-4 win over former world No.1 and 2021 US Open and Canadian Open champ Daniil Medvedev of Russia.
Popyrin has his sights set on back-to-back Canadian Open titles after thrashing Russia's Andrey Rublev 6-2 6-4 in last year's final.
On Saturday he took some time to immerse himself into the match against Rune.
He was frustrated not to cash in on any of the 11 break points he had in the first set.
However, he turned things around after that, converting both of his break points in the second set, then doing the same with his first opportunity in the final stanza.
"I started this week not high on confidence, needing to get a grasp of the mental side,'' the Australian said.
"After the first set I was pissed (about not breaking), but when I came back out to return again it felt like I was in control of the match.
"I told myself to be aggressive with the next break point that I got, and from them on I felt quite comfortable.''
In the quarter-finals, Popyrin faces top-seeded Alexander Zverev, who advanced after the No.14 seed Francisco Cerundolo, of Argentina, was forced to retire with an abdominal injury with the German leading 6-4 1-0.
Zverev, the 2017 Canadian Open champion who is ranked third in the world, ended up the top seed in Toronto after the world's two top-ranked players Jannik Sinner and Carlos Alcaraz skipped the event.
Another Australian, ninth-seeded Alex de Minaur, plays his round-of-16 clash against American Frances Tiafoe on Sunday (Monday AEST).
De Minaur came to Canada in impeccable form, having won last week's Washington Open, his 10th career title.
The fleet-footed Aussie will be hoping to continue his dazzling recent form at the year's final major, the US Open, which starts in New York on August 24.
De Minaur has twice reached the quarter-finals at the hard-court event at Flushing Meadows, firstly in 2021, then repeating that effort last year.
In other results on Saturday, Alex Michelsen reached the quarter-finals after toppling fellow American Learner Tien 6-3 6-3.
Michelsen next faces 11th-seeded Russian Karen Khachanov, a 6-4 7-5 winner over eighth-seeded Norwegian Casper Ruud.
- with AP
6°C