The 26-year-old Kazakh, who was champion here in 2024 but skipped last year's event, relied on her service game as she won 6-2, 6-4 against the unseeded Russian in one hour and 12 minutes.
Shnaider, ranked 19th in the world, improved in the second set but was undone by double faults at crucial moments, allowing Rybakina to capitalise.
Rybakina capped the match off with her ninth ace and will next take on either Leylah Fernandez or Zeynep Sonmez in the quarters.
Earlier, world number nine Mirra Andreeva pushed past American Alycia Parks in straight sets in their last 16 clash to extend her winning streak to six matches.
Sixth seed Andreeva is in fine form, having won the clay event in Linz last week. She eliminated defending champion Jelena Ostapenko on Wednesday and breezed past Parks 7-6(3), 6-3.
The Russian will take on two-time Stuttgart winner Iga Swiatek, who won her first clay court match of the season with a comprehensive victory over Laura Siegemund on Wednesday.
"Obviously, she's a great player, but I'll just approach this match as every other match that I've had before," the 18-year-old Andreeva said of her quarter-final opponent.
"She's had a good history on clay, so I'm just very curious to see how it's going to go."
Czech Karolina Muchova came from a set down to beat Elise Mertens 1-6, 6-3, 6-0 to book her place in the last eight.
Muchova's compatriot Linda Noskova crushed Russian Ekaterina Alexandrova 6-1, 6-1 to set up a quarter-final clash with Elina Svitolina, who cruised past German Eva Lys 6-1, 6-0 on Wednesday.
Later on Thursday, world number three Coco Gauff starts her bid for a first Stuttgart title when she takes on Liudmila Samsonova.