The Austrian won 74 percent of first serve points compared to just 60 percent for Auger-Aliassime.
De Minaur, however, then suffered a mini-meltdown, losing eight games in a row, allowing Khachanov back into the match as the Russian opened up a 2-1 lead.
When the US Open started neither promising 21-year-old De Minaur nor 30-year-old Pospisil, who grabbed more attention in recent weeks as Novak Djokovic's sidekick in starting a new players association, would have been tipped as a major victor. "I'm happy I'm finally at this stage".
The 15th seed has not dropped a set since the first round, when Thiago Monteiro took the second in their four-set clash.
"I played aggressive, I executed extremely well. That was kind of my mentality", he added.
He is a 21-year-old from Australia who is seeded 21st at Flushing Meadows. Then took it up a notch pounding winners from the baseline against the talented Canadian who was appearing in his first Grand Slam fourth round. "I want people in the locker-room to know that if they're going to beat me, they're going to have to go through me".
Aliassime cruised through the first two sets before Moutet battled in the third.
Serena Williams has now won back-to-back three-set matches
Thiem and Auger-Aliassime are both strong movers and very good athletes overall, meaning this could be quite a physical contest at times.
"I'm playing good. I'm playing better every match".
Auger-Aliassime will next face the victor of a match between No. 2 seed Dominic Thiem of Austria and No. 31 seed Marin Cilic of Croatia.
De Minaur, 21, kept his composure to save four set points in the first set tiebreak and although Pospisil fired 43 winners in the match to De Minaur's 16, the Canadian also had 48 unforced errors. "With the way I served, I was only broken once, which didn't really have a huge impact on the score at the end".
Returning at 5-1, Thiem landed a backhand down the line victor for the third break of serve at love and the place in the last eight.
Meanwhile, third-seeded Daniil Medvedev swept aside Frances Tiafoe 6-4 6-1 6-0 in the late match to power into the quarters.
Andrey Rublev serves to Matteo Berrettini during the fourth round of the US Open on Monday. "Whether it's tennis or whatever, I just want to see people of colour in my communities win". In the evening Pablo Carreno Busta looks to return to the US Open semifinals against Canadian lefty Denis Shapovalov who is appearing in his first Grand Slam quarterfinal.