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De Minaur overcame a mid-match meltdown to progress to the last 16 for the second straight year with a spirited 6-4 0-6 4-6 6-3 6-1 victory over the 11th seed on Sunday.
The 21st seed felt compelled to apologise for losing his cool and destroying a racquet after dropping eight games in a row to surrender the third set, but was otherwise delighted to advance to the second week of a slam for only the second time.
"I'm happy that I got through it and I managed to kind of turn things around," de Minaur said.
"But it's something that just can't happen and won't happen again.
"Look, it was a tough one. Happy to be in the second week of a major again."
De Minaur next plays big-hitting Canadian Vasek Pospisil on Tuesday for a place in the quarter-finals.
Pospisil also recovered from two sets to one down on Sunday to eliminate Spanish eighth seed Roberto Bautista Agut 7-5 2-6 4-6 6-3 6-2 and blow the bottom half of the draw wide open.
"Well, obviously even though Vasek isn't a seed, he's definitely playing like a top seed," de Minaur said.
"He's taken out both Milos (Raonic) and Bautista. He definitely deserves to be there. He's a very, very dangerous player."
With de Minaur's Sydney hitting partner Jordan Thompson already through, it's the first time two Australian men have made the second week at Flushing Meadows since former champions Pat Rafter and Lleyton Hewitt did so in 2001.
De Minaur was staring down the barrel after Khachanov powered through the second and third sets in quick fashion.
But the 21-year-old refused to say die, levelling the match, then breaking the Russian's spirit in the deciding set.
Driven mad by de Minaur's relentless counter-punching and unrivalled foot speed, Khachanov double-faulted on game point at 1-1 in the deciding set.
The dispirited Russian then dumped a backhand volley into the net after de Minaur rifled a brilliant running forehand to seize a decisive second break for a 4-1 lead.
With eight aces and 42 winners, de Minaur sealed victory after three hours and two minutes, emulating countryman Nick Kyrgios' five-set third-round triumph over Khachanov at the Australian Open in January.
Both de Minaur and Thompson are eyeing maiden grand-slam quarter-final berths at Flushing Meadows.
Thompson faces No.27 seed Borna Coric on Monday after the Croatian saved six match points to oust Greek fourth seed Stefanos Tsitsipas in a five-set fourth-round thriller on day five.
