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The Spaniard, 31, triumphed in an hour 38 minutes against 21-year-old world number 38 Khachanov.
Austrian fifth seed Dominic Thiem beat Novak Djokovic to set up a quarter-final against Nadal on Friday.
Former world number one Djokovic lost 6-7 (2-7) 6-2 6-3 in an engrossing match.
The Serb has not won a Grand Slam title since the 2016 French Open and has failed to reach the last eight in any of his four tournament appearances since making his return following an elbow injury.
"Ideally, I'd want to be playing the way I played for so many years, but obviously patience is required," said Djokovic, who confirmed he will play in either Budapest or Barcelona next week as a wildcard.
"I really believe if I continue doing what I'm doing, believing in myself, I know eventually I'm going to get there."
Djokovic, who has slipped to 13th in the rankings, showed glimpses of his best form against Thiem, whom he described as a "top-three player in the world on clay".
The 12-time Grand Slam champion saved three set points at 5-2 down in the first, breaking back to level at 5-5 and taking the tie-break with his first set point.
However, he was outclassed in the second set and seemed to fade in the decider, Thiem claiming his first break point for 4-3 and taking Djokovic's serve again to win after almost two and a half hours.
Thiem, 24, lost his first five meetings with Djokovic before beating him 7-6 (7-5) 6-3 6-0 in last year's French Open.
Nadal, who must win the tournament to prevent Roger Federer from regaining the number one spot, eased past Khachanov to claim his 65th victory at a tournament where he has lost only four matches.
The Spaniard broke serve at the first attempt and, although he lost his own serve when leading 4-2 in the first set, he broke again in the next game before serving out the set.
Nadal, who is bidding for a 17th Grand Slam title at next month's French Open, raced into a 4-1 lead in the second set and wrapped up victory on his first match point.
"It was a good victory - straight sets against a difficult opponent. That's the real thing," he said.
Elsewhere, third seed Alexander Zverev beat fellow German Jan-Lennard Struff 6-4 4-6 6-4, but missed out on a meeting with his older brother Mischa after he lost to France's Richard Gasquet.
Gasquet, 31, earned his 500th Tour victory with a 6-2 7-5 win.
Belgian sixth seed David Goffin won 6-4 7-5 against Spain's Roberto Bautista Agut.
Bautista Agut was given a code violation for accidently hitting a ball boy with a ball after going 4-1 down in the second set, recovering to save two match points and winning four straight games to level at 5-5.
But Goffin broke again and served out the match at the third attempt to book a quarter-final against Bulgarian fourth seed Grigor Dimitrov, who beat Germany's Philipp Kohlschreiber 4-6 6-3 6-4.
Canada's Milos Raonic pulled out of his last-16 match against Croatian second seed Marin Cilic because of a knee injury.
(The Guardian)
