The 35-year-old Serb is aiming to become the oldest men's singles champion in Paris in the Open era as he chases a record-equalling 21st Grand Slam title.
Next up for Djokovic, who was also French Open champion in 2016, is a match-up with Argentina's Diego Schwartzman, the 15th seed.
Nadal, the 21-time major champion, improved his record at Roland Garros to 108-3 after seeing off 2021 US Open quarter-finalist Van de Zandschulp.
He has now reached at least the fourth round in 17 of 18 visits to Roland Garros, with the exception of 2016 when he withdrew in the third round with a wrist injury.
"Today has been a very good test and very positive test, because I was able to play my best match of the tournament so far, without any doubt," said Nadal.
The Spaniard had a break point for 5-0 in the third set, but Van de Zandschulp refused to roll over, winning four of the next five games before Nadal put away his second match point, having saved a break point in between the two.
"You always feel a little bit the nerves to finish the match, so I'm happy to be through in straight sets," said Nadal, who plays Canadian ninth seed Felix Auger-Aliassime for a place in the last eight.
In a twist to that clash, Toni Nadal, the uncle and former coach of the Spaniard, is now working with Auger-Aliassime.
Alcaraz, Zverev move on
Alcaraz gained revenge against the only man to beat him on clay this year, defeating US 27th seed Sebastian Korda 6-4, 6-4, 6-2 in the evening session.
The Spanish teenage star is the youngest man in the Roland Garros fourth round since Djokovic 16 years ago.
"I'm trying to have fun out there. I just enjoy every single second. I love playing tennis," said Alcaraz, who saved a match point against countryman Albert Ramos-Vinolas in the previous round.
He has won 20 of 21 matches on clay this season and next faces 21st seed Karen Khachanov. The Russian knocked out Cameron Norrie in four sets as the 10th-seeded Briton became the highest-ranked player to fall.
Alexander Zverev, the third seed and 2021 semi-finalist, ended the run of American Brandon Nakashima, winning 7-6 (7/2), 6-3, 7-6 (7/5).
The German Olympic champion, who also saved a match point in round two, next takes on Bernabe Zapata Miralles, the 131st-ranked Spaniard who downed John Isner in five sets.
Amanda Anisimova, the 27th seed and a 2019 semi-finalist, advanced to the last 16 as injury-plagued Karolina Muchova retired down 6-7 (7/9), 6-2, 3-0 after falling and twisting her ankle midway through the second set.
Last year's US Open runner-up Leylah Fernandez beat Olympic champion Belinda Bencic of Switzerland 7-5, 3-6, 7-5, while Aliaksandra Sasnovich defeated three-time Grand Slam winner Angelique Kerber 6-4, 7-6 (7/5).
Coco Gauff, the youngest player left in the French Open, made the last 16 with a straight-sets win over Kaia Kanepi, the oldest woman in the tournament at 36.
Sloane Stephens, runner-up in 2018, ended the run of French teenager Diane Parry in straight sets, while Swiss 23rd seed Jil Teichmann overcame two-time former Australian Open winner Victoria Azarenka 4-6, 7-5, 7-6 (10/5).