It seemed that Zverev would ease through as he led by a set and a break within an hour, holding a point for a double break, though Dzumhur sensationally turned the match around and served for the match in the fourth set, only to Zverev to win a tiebreak.
But the 31st-seeded Buzarnescu, a Romanian with a PhD in sports science, continued the surge that has seen her rise almost 350 spots in the rankings over the past year.
Zverev played with a lack of consistency against Dzumhur for much of the match with 73 unforced errors, dropping serve eight times and serving seven double faults.
He was a first-round casualty in Paris past year and nearly came a cropper in the second round on Wednesday when he had to fight back from two sets to one down to beat Dusan Lajovic.
World number three Alexander Zverev came through other five-set encounter to reach the fourth round at the French Open on Friday, with the German saving a match point to beat 26th seed Damir Dzumhur in one of the matches of the tournament so far. But he picked himself up with Dzumhur's help, and play resumed.
Kei Nishikori advanced to the fourth round of the French Open by beating Gilles Simon 6-3, 6-1, 6-3.
Bulgaria's Grigor Dimitrov reacts as he plays Spain's Fernando Verdasco during their third round match of the French Open tennis tournament at the Roland Garros stadium, Friday, June 1, 2018 in Paris.
Newcomer wins National Spelling Bee after 4-time participant botches 'Bewusstseinslage'
Remmers said her daughter, a seventh-grader at Target Range School, was watching her face when she was on the phone. The most commonly misspelled word in the United States, carry the number one spot in eleven states, is "beautiful".
But he has never reached a Grand Slam quarter-final, although the longer best-of-five format has worked twice in his favour now at Roland Garros.
Dimitrov, a two-time major semifinalist, bowed out 7-6 (4), 6-2, 6-4 in the third round against Fernando Verdasco, who is ranked 35th and seeded 30th.
The set was decided by a tiebreaker which the Spaniard won 8-6 to tie the match up at a set apiece.
Verdasco, meanwhile, got to the fourth round at Roland Garros for the seventh time. Osaka, seeded 21, is bidding to reach the fourth round in Paris for the first time to match her run to the last-16 in Australia in January. He also had to save a match point in the 11th game of the decider before breaking to punch his ticket for the final 16. Dzumhur served out the set 6-3.
Dzumhur was a man on a mission in an epic third-round contest, giving both Alexander Zverev and a ball boy an almighty scare before he was beaten on Court Philippe Chatrier. It doesn't matter if it goes 9-7 in the fifth or it goes 6-1, 6-1, 6-2.
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.
World number two and Australian Open champion Caroline Wozniacki continued her impressive run in Paris with a 6-0 6-3 win over local hope Paul Parmentier in the third round.