Former Wimbledon champion Rybakina reached the fourth round for the second time with a commanding win over Elise Mertens.
The world number four needed just 67 minutes to beat the 25th seed from Belgium 6-4 6-2.
Rybakina, 24, reached the quarter-finals in Paris in 2021 but withdrew with injury before her third-round match last year.
The in-form Kazakh will face Elina Svitolina after last year’s Wimbledon semi-finalist defeated Romania’s Ana Bogdan 7-5 6-2.
Earlier, home hope Varvara Gracheva overcame Romania’s Irina-Camelia Begu 7-5 6-3 to reach the fourth round of a major for the first time.
The Russia-born player, ranked 88th in the world, was serenaded by the crowd on Court Suzanne Lenglen with a loud rendition of the national anthem, La Marseillaise.
“I will remember this moment until the end of my life,” Gracheva, who has represented France since 2023 after living in the country for more than six years, said.
“It means that everyone accepts me, that I’m home here.”
She will face Russian teenager Mirra Andreeva after her convincing 6-2 6-1 win over American Peyton Stearns.
Meanwhile, Chinese seventh seed Zheng could not close out a remarkable post-rain delay comeback against Russia’s Avanesyan.
Zheng was a break down in the third set when play resumed on Saturday afternoon and she conceded serve again before fighting back from 5-2 down to force a tie-break.
But Avanesyan took control late in the breaker, winning four points in a row to secure her spot in the fourth round for a second straight year in Paris.
She will take on 12th seed Jasmine Paolini in round four, after the Italian beat former US Open winner Bianca Andreescu 6-1 3-6 6-0.