Australian Open champion Rybakina beat Diana Shnaider 6-2, 6-4, while Gauff overpowered Liudmila Samsonova 7-5, 6-1, after Andreeva beat Alycia Parks 7-6(3), 6-3.
World number two Rybakina put on a clinical display, never facing a break point on her own serve and landing nine aces, while Shnaider could manage only one.
"Happy with the win, definitely, but still a lot to improve," Kazakhstan's Rybakina said after her first claycourt match of the year.
American Gauff, also playing her first match on clay this season, got broken twice as Samsonova took a 3-0 lead in the first set, but the world number three bounced back to win the next four games. Samsonova broke again to go 5-4 up, but Gauff won the next three games to win the set.
A confident Gauff won five games straight in the second set to set up a last-eight clash with Karolina Muchova.
"It's a surface where I love and hate it," Gauff said about playing on clay.
"It's fun when you hit some good ones, but also not fun when you're falling on the ground. But today I didn't fall, so that's a plus."
Russian world number nine Andreeva, who won the Linz Open at the weekend before knocking out the Stuttgart Open defending champion Jelena Ostapenko on Wednesday, beat American Parks to win her sixth straight claycourt match this year.
Parks led 6-5 before the 18-year-old Andreeva fought back to win the first set in a tiebreak. The American, who had seven double faults in the match to Andreeva's two, also went a break up in the second set but was unable to hold onto the lead.
Polish third seed Swiatek, who is this year aiming for a fifth French Open title, began her clay season with a 6-2 6-3 win over Laura Siegemund in the last 16 on Wednesday.
Andreeva has won two of her three previous meetings with Swiatek, all of which took place on hardcourt.
"Obviously, this is clay indoors, so it's still a little bit different, but yeah, I'm kind of excited to see how this match will go on clay court for the first time," Andreeva said about Friday's clash.
