Auger-Aliassime retrieved a break in the first set before seizing control against the 15th-ranked Rublev, triumphing 7-5, 6-3, 6-4 for just his second win over the Russian in nine attempts.
The 25-year-old Auger-Aliassime backed up his win over third seed Alexander Zverev in the previous round and will play Australia's Alex de Minaur in the last eight.
Auger-Aliassime climbed as high as sixth in the world at the end of 2022 but had not made it to a major quarter-final since that year's Australian Open.
His deepest run at a Grand Slam came when he advanced to the semi-finals of the US Open four years ago.
"It feels even better than the first time," Auger-Aliassime said of returning to the last eight in New York.
"I think the first time at 21, I was kind of on my way up. To have a few setbacks, injuries, struggles with confidence... to come back for a second time to the quarter-finals here, it feels much better.
"It feels more deserved. I'm soaking in every moment here."