The speedy Australian fell 5-2 behind in the opener and had to fend off three set points before clinching the first set against the 2021 champion in Toronto.
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De Minaur had to rally from a break down twice in the second set before showing his resilience again thanks to his brilliant return game, sealing victory on the Russian's seventh double fault.
Up next is a meeting with red-hot Spaniard Alejandro Davidovich Fokina, who swept aside American Mackenzie McDonald (28) 6-4, 6-2 earlier on Friday.
Davidovich Fokina holds a 2-1 lead over De Minaur in their head-to-head meetings.