The match will resume Monday. Stephens will face No. 14 Daria Kasatkina of Russian Federation, and Keys faces 98th-ranked Yulia Putintseva of Kazakhstan.
The clay courts of Roland Garros host the second Grand Slam of the season.
"It was an unbelievable, high level match, it was the first time I have won on this court and I hope there are many more to come", said Zverev, bidding to become the first German men's champion in Paris since 1937.
The 2016 champion beat 13th-seeded Roberto Bautista Agut of Spain 6-4, 6-7 (6), 7-6 (4), 6-2, in a near four-hour match too sluggish for Djokovic's liking.
Serena Williams gave another airing to the catsuit as she and sister Venus made it through to the third round of the women's doubles.
Stephens, the U.S. Open champion in 2017, blew past Anett Kontaveit of Estonia 6-2, 6-0 in just 52 minutes on Court Philippe Chatrier.
Djokovic, seeded 20th, becomes the only player to have reached the last 16 of the French Open at each of the last nine editions.
The match was made even more dramatic when Bosnian-born Dzumhur collided with a ball boy who lay motionless on the ground after the pair both went after the same high ball.
In the fifth set, Dzumhur held match point with Zverev serving down 4-5, but the German pulled out a fine first serve, hitting the line and forcing Dzumhur wide on his backhand.
Thiem bit his fingers after squandering his first match point.
Djokovic won on his second match point with an easy smash at the net.
The ATP has promoted its young guns through their "NextGen" series since the start of last season, playing a "NextGen" Finals in Milan previous year for the best eight players aged 21 and under. He'll play the second-seeded Alexander Zverev next.
Unseeded Italian Marco Cecchinato also kept up his strong play, beating 10th-seeded Spaniard Pablo Carreno Busta 2-6, 7-6 (5), 6-3, 6-1.
The 21-year-old broke Dzumhur in the 11th game and then served out the match to secure his first Grand Slam win against a top-50 opponent at the eighth time of trying.
In contrast, Thiem has reached the last eight at Roland Garros for a third consecutive year - which will make for a thrilling tie.
In his previous match, Dimitrov had come back after trailing two sets to one and defeated Jared Donaldson 10-8 in the fifth set.
BC-TEN-French Open, 2nd Ld-Writethru, 6225-set survivor Zverev living up to French Open billingAP Photo PDJ200, PDJ219, PDJ218, PDJ154, PDJ136, PDJ165, PDJ184Eds: Updates with wins for Djokovic, Verdasco and Wozniacki, and Zverev quotes, details.
He might well need to up that schedule considering his quarter-final opponent is No. 7 Dominic Thiem of Austria, who is one of the fittest players on the men's circuit and has reached the past two semifinals at Roland Garros.
Her feat means the 13th-seeded American has now reached the last eight at every major.
"Trust me, in that moment when I do it, I don't think about how it can help me", the smiling Serbian said when asked about the incident. "Hopefully big things can happen here".
The 28-year-old saw off Simon 6-3, 6-1, 6-3.
Djokovic's desire has been questioned during his fall from the sport's pinnacle but he was certainly fired up on Court Suzanne Lenglen.
The victory was his first on Court Philippe Chatrier. The German, however, was able to break straight back, though wobbled later on, allowing Dzumhur to serve for the match at 6-5 in the fourth.
Serving a second time for the match, the 23-year-old Keys let out a scream of "Come on!" when she hit a superb cross-court backhand victor to make it 40-15 and give herself another two chances for victory.
China: Smaller trade surplus, OK. But tech concessions? No.
Trump said last month that he was working with Chinese President Xi Jinping to give ZTE "a way to get back into business, fast". On Sunday, Ross said the two sides discussed specific exports Beijing might buy, but the meeting ended with no commitments.
Trudeau: Trump's tariffs are 'insulting and unacceptable'
Prime Minister Justin Trudeau has taken exception to the fact that the USA has imposed tariffs for "national security" reasons. The administration is also placing tariffs on $50 billion of various Chinese goods, citing unfair Chinese trade practices.