Verdasco was twice a break up in the third set, only to see Dimitrov fight back on both occasions, but eventually broke again in the 10th game, converting his first match point to set up a fourth-round match with either Novak Djokovic or Roberto Bautista Agut.
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.
"We know each other pretty well", the 21-year-old Zverev said.
She has not dropped a set yet and will face U.S. Open runner-up Madison Keys in the fourth round.
He was fortunate to survive a break point when serving for the set at 5-3, with Khachanov's blazing forehand clipping the net cord and bouncing just millimetres behind the baseline, before drawing level at two sets apiece with an ace. But he then picked himself up with Dzumhur's help.
Novak Djokovic equaled Roger Federer's mark of reaching nine consecutive quarterfinals at Roland Garros by beating No. 30 seed Fernando Verdasco 6-3, 6-4, 6-2.
Nishikori has twice reached the quarterfinals at Roland Garros, including previous year.
First lady to hang back on president's Camp David weekend
She was last spotted on May 10, when she joined her husband at Andrews Air Force Base to welcome home hostages from North Korea . She addressed them on Wednesday, tweeting: "I see the media is working overtime speculating where I am & what I'm doing".
His favoured shot is his backhand, which he can hit with power both cross-court and down the line.
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. That 4-hour, 19-minute victory might have taken something out of Dimitrov. 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. Dzumhur served out the set 6-3.
Bosnia's Damir Dzumhur had an unfortunate run-in with a ball boy Friday at the French Open. However, it was the Bosnian who continued to look the bigger threat, and after he broke to love at 4-4, calmly served out the first set to edge nearer to victory. "Sometimes you have to battle to get the win and that's what I did today".
He then cruised through the final set winning it 6-2.
Zverev, 21, who is seen as Rafael Nadal's principal obstacle to the Spaniard's expected coronation as champion for an 11th time, endured a nightmare outing before claiming victory after nearly four hours on Court Philippe Chatrier.
Dimitrov is playing left-hander Fernando Verdasco on Court 1.
In the women's draw, second-seeded Caroline Wozniacki breezed past French wild-card entry Pauline Parmentier 6-0, 6-3 and appears to be brimming with self-belief after her breakthrough Australian Open title in January.
Spanish 13th seed Bautista Agut was beaten by the former champion in four sets, just a week after his mother Ester passed away.