Nadal, who celebrated his 31st birthday over the weekend, has conceded just 20 games in four matches to secure his place in the last eight.
With the win, Nadal marched into the quarter-finals for the 11th time, thus joining 18-time Grand Slam champion Roger Federer at the top of the chart for most Roland Garros last-eight showings in the Open Era.
"Everything wasn't flawless. There were a few concerns", Mladenovic said, before joking: "I made about 35 double-faults today, but everything's OK!"
"I have obviously played him a year ago [in the] semis here, played a really good match, played an incredible match against him in Rome. I'm feeling like I'm missing consistency", Djokovic said.
"He is a one of the best players in the world, especially on this surface".
Halep, who was runner-up at Roland Garros in 2014, will next face second-seeded Karolina Pliskova.
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Things did not begin promisingly, but after twice fighting back from a break down in the opening set he pulled away to win 7-6 (7/5) 6-1 6-3.
At 21, Khachanov is the youngest man to reach the round of 16 at Roland Garros since US Open victor Marin Cilic, then 20, in 2009.
Simona Halep and Karolina Pliskova reached the women's semifinals. That happened less than an hour after defending champion Garbine Muguruza was beaten 6-1, 3-6, 6-3 by 13th-seeded Kristina Mladenovic at Court Suzanne Lenglen.
Chung forced the contest into a deciding set as play resumed on Sunday, but the world number 67 double faulted on match point to gift Nishikori a 7-5, 6-4, 6-7 (4/7), 0-6, 6-4 victory.
The Serbian virtually gave up in the third set in what was a shocking end to his title defence.
Novak Djokovic and other players at the French Open say they are not anxious about competing at Wimbledon next month despite the recent attacks in Britain. Nadal advanced when his opponent, Pablo Carreno Busta, stopped in the second set because of an injured abdominal muscle.
Kuznetsova, who won the 2009 French Open and 2004 U.S. Open, was ousted by two-time major finalist Caroline Wozniacki 6-1, 4-6, 6-2, while 2011 U.S. Open champion Stosur was eliminated by 19-year-old Jelena Ostapenko of Latvia 2-6, 6-2, 6-4.