Djokovic, the 2016 French champion and 12-times Grand Slam victor, awaits.
Khachanov, the 6ft 6in Russian ranked 38 in the world, this time had no answer as Zverev took the match 4-6 7-6 (7/4) 2-6 6-3 6-3.
But Zverev, a powerful 21-year-old with a complete game, had never reached a Grand Slam quarterfinal until now.
After fighting through three consecutive five-set match-wins, Zverev is still on track to be the first German men's champion at Roland Garros since 1937.
Zverev has won eight titles, including three Masters 1000 crowns, at the tour level but his abilities to deliver at crunch moments and remain unaffected by fatigue were questioned following his dismal performances in Grand Slams in the past.
World number three Alexander Zverev expressed confidence that youngsters on the tour can break the dominance of Rafael Nadal and Roger Federer, the two veterans of the sport who are still atop the ATP rankings. I think in the end we'll be very high soon. "So it's going to be an incredible challenge for me".
Novak Djokovic strengthened his claims as a French Open contender by reaching the quarter-finals with a straight-set victory over Spain's Fernando Verdasco.
The unheralded Italian upset Belgian seventh seed David Goffin 7-5 4-6 6-0 6-3 to qualify for his first grand slam quarter-final.
The German, who has previously stated his desire to be the best in the men's game before Nadal and Federer - now ranked number one and two in the world respectively - retire, once again stated his intention to do so during the French Open that is underway in Paris.
No. 10 Sloane Stephens vs.
Stephens, the 10th seed, had previously made the fourth round for four successive seasons and will next play either Australian Open champion and second seed Caroline Wozniacki or Russia's Daria Kasatkina. Hopefully it's going to get even better.
Verdasco made 48 unforced errors and grew increasingly furious with himself as he missed out on a first French Open quarter-final at the 16th attempt.
In the women's quarterfinals, two Americans could win and set up a semifinal rematch of their U.S. Open final a year ago: No. 10 Sloane Stephens and No. 13 Madison Keys.
"First week, my goal is to get to the second week", said 23-year-old Keys.
But all eyes will be on Monday's showdown between Serena Williams and Maria Sharapova, with the latter trying to end an 18-match losing streak to her bitter rival.