Djokovic, chasing a third Roland Garros championship and 23rd men’s Grand Slam title, eased past 94th-ranked Peruvian opponent Juan Pablo Varillas, 6-3, 6-2, 6-2.
The 36-year-old Serb, champion in Paris in 2016 and 2021, will face 11th seed Karen Khachanov for a place in the semi-finals.
Djokovic is one of the favourites to claim the French Open this year with 14-time champion Nadal sidelined for the first time since 2004 after failing to recover from a hip injury.
On Saturday, the day of his 37th birthday, it was revealed that Nadal would require at least five months of rehabilitation following keyhole surgery on his injured hip.
That means he will likely not play again until 2024 which will be his final season on tour.
“I really hope that his rehabilitation process can go well and that we can see him next season,” said Djokovic of Nadal who he has faced 59 times in his career.
He and the Spaniard are tied on 22 Grand Slam titles.
“He’s so important for our game on and off the court, one of the greatest legends of tennis. We want to see healthy Rafa, no question about it, playing for what he has announced his last season. Hopefully he’s gonna be able to do that.”
On his record 17th quarter-final in Paris, and 14th in a row, Djokovic added: “I am very proud of this record. I have put a lot of effort into my game and I am very motivated to continue.”
“However, my attention is already on the next match. I know what my goal is here. I’m trying to stay mentally the course and not look too far.”
Djokovic holds a commanding 8-1 career lead over Khachanov, including their only previous meeting at the French Open in 2020.
The Russian reached the quarter-finals for the second time with a 1-6, 6-4, 7-6 (9/7), 6-1 win over Lorenzo Sonego of Italy.
The 27-year-old Khachanov says he hopes his fondness for chess will help his strategic and tactical battle of wills with Djokovic on Tuesday.
“You need to play really good chess to beat him,” said Khachanov.
“I like to play chess in the morning to start the brain working and going. I think there are some similarities here.
“I think when you open your head and you are really full into the game, you see the field much bigger. You see the possible shots which you can make troubles to the opponent. I think there are some similarities. Maybe it helps me.”