The four-time Paris champion faces Marin Cilic in the last eight on Friday.
Khachanov beat John Isner 6-4, 6-7 (9), 7-6 (8) in a match in which the eighth-seeded Isner had 19 aces but missed two match points. Djokovic next faces Damir Dzumhur.
Roger Federer also advanced to Thursday's third round after big-serving Milos Raonic retired before their match with a right elbow injury.
Federer's only tournament victory in Bercy came in 2011. It's been a tough year for me with injuries. He has also sat out the last three French Opens.
"I always believe in myself, but five months ago, if you'd told me I would be back to number one so quickly, it was highly improbable considering my ranking and the way I played and felt on the court".
If he wins that, against Kei Nishikori, he could face Novak Djokovic in the semifinals - but Federer is thinking more about winning the season-ending ATP Finals, which start November 11. "I would love to be in London, of course", said Nadal, who will wait on making a decision on his participation at the season finale.More news: Retina MacBook Air may be Apple’s most handsome laptop ever
"I feel like I've recovered well after last week". So yeah, I'll give it a go tomorrow.
Raonic fired 23 aces as he outlasted France's Jo-Wilfried Tsonga 6-7 (4/7), 7-6 (7/5), 7-6 (7/5) in his opening match.
The 31-year-old has put himself in a position where he can leapfrog Nadal and finish the year as the top-ranked men's singles player. "I checked with the doctor and the doctor recommended I did not play".
"It will not be fair to say it's a real injury today but what is sure that if I continue then it will be a real injury".
Nadal and Djokovic are scheduled to meet in Jeddah on Dec 22, but they have faced growing calls to withdraw following the murder of a Saudi journalist in the country's consulate in Istanbul.
Meanwhile, Djokovic will become the new world No 1 on Monday when the latest rankings are announced.
The fifth seed bounced back from a bruising defeat to Anett Kontaveit on Wednesday, but overcame Mertens 6-2, 7-6 in a tense round-robin tie in Zhuhai, southern China.