Last year's US Open saw Draper leading Khachanov by a break in the third set of their third-round clash, before having to retire from the match due to injury.
However, on this occasion the 21-year-old Next Gen ATP Finals competitor was unable to register a break against the Russian, who won 6-4 6-2 and will now face compatriot Daniil Medvedev.
Khachanov said on atptour.com: "To be honest I was expecting a very tough match against Jack because we had a good battle in the US Open, which unfortunately finished not in the way I wanted, but he retired in the end of the third set.
"He's on a good run, he's a good rising-up player, younger than me. To be honest with you from the beginning until the end, I played a very solid match, very good performance from my side."
Medvedev wasted little time in cruising past Miomir Kecmanovic 6-0, 6-3 to book his place in the quarter-finals.
"It's always better to win 6-0 than to win 7-6 saving nine set points and playing one hour, 20 [minutes]," Medvedev said. "But tennis is not as easy, otherwise everyone would be No. 1, but we have only one. So happy with my level today against a strong opponent. Second set was tougher, but managed to close it out at the end and looking forward to my next matches."
Medvedev remains on a collision course with Novak Djokovic, should both men advance to the semi-finals.
However, the Russian is not ready to look past Khachanov at this stage.
He's a very strong opponent, he can beat anybody," Medvedev said. "He beat Jack today, who is a young guy on the rise, so looking forward to this tough match."