Handed a bye for the opening round, third seed Khachanov made home advantage count in his first match as the Russian defeated Australian world number 50 James Duckworth 3-6 6-3 6-1 to reach the quarter-finals on Thursday (AEDT).
Khachanov – Olympic Games silver medallist, who triumphed in Moscow three years ago – will face another Australian, John Millman after he defeated Illya Marchenko 6-1 5-7 6-4 at the ATP 250 tournament.
"It always feel great to be back in your hometown, in a tournament where everything started for me as a professional tennis player," world number 31 Khachanov said.
"[This is] where I won my first matches, I got my first points. Since then, it's always great to come back home to see a crowd, [get] home support. It's also little bit more nervous, of course. But I always enjoy playing at home."
Khachanov was joined in going through to the last eight by sixth seed Marin Cilic – a two-time Kremlin Cup champion who overcame Tommy Paul 7-5 6-3.
At the European Open in Antwerp, top seed Jannik Sinner downed fellow Italian youngster Lorenzo Musetti 7-5 6-2 in the duo's first meeting as professionals.
"We never practice together, so it was kind of a new match," Sinner said post-match.
"He's an incredible player. He's a very great talent, so I hope we will have a couple of more matches on the pro tour, and obviously I'm very happy about my level today."
Arthur Rinderknech upstaged eighth seed Dusan Lajovic 6-3 4-6 6-2 to set up a quarter-final tie with Sinner, who is in the hunt for a fourth title of the season and a place at next month's ATP Finals in Turin.
Another emerging talent, qualifier Jenson Brooksby, bettered US Open quarter-finalist Botic van de Zandschulp 6-2 6-0, with Alejandro Davidovich Fokina sending third seed Cristian Garin packing 4-6 6-3 6-3.