Shnaider captured her first match against a top-10 rival in winning the meeting of 20-year-olds with world number two Gauff, the US top seed who was ousted in a bid for her eighth career WTA title.
"I tried to play it cool, but inside I was like, 'Yes, I made it," said Shnaider.
Asked how she pulled off the stunner, Shnaider replied, "Honestly I still don't know. It was a little bit tough conditions. It was very windy but I'm very glad I managed to overcome it."
The 24th-ranked Russian seeks her fourth WTA title of the year after Thailand, Bad Homburg and Budapest.
Shnaider, in her first WTA Masters level round of 16 match, needed only 70 minutes to dispatch Gauff and book a quarter-final against Russian sixth seed Liudmila Samsonova, who eliminated Belgium's Elise Mertens 6-2, 6-4.
Shnaider broke Gauff in the 10th game to capture the opening set and the American double-faulted away a break to hand Shnaider a 3-1 lead in the second set and never recovered.
"Sometimes I question like whether I should have played or not," Gauff said. "But at the end of the day I wanted to test myself and see if I would be able to, how I would do being mentally tired a little bit and physically fatigued.
"I didn't have high expectations, but I wish I could have competed better today, even if it resulted in a loss. I don't think I competed well."
Gauff will play next week at Cincinnati, where she is the defending champion, in the final major tune-up for the US Open.
US third seed and defending champion Jessica Pegula defeated compatriot Ashlyn Krueger 6-2, 6-4 to reach a quarter-final against another American, Peyton Stearns, who advanced 6-4, 4-2 when 12th seed Victoria Azarenka retired.
Another all-US quarter-final will send eighth seed Emma Navarro against Taylor Townsend.
Navarro defeated Ukraine's 11th seed Marta Kostyuk 7-5, 7-5 while Townsend upset Latvian fourth seed Jelena Ostapenko 6-2, 6-1.
Two-time reigning Australian Open champion Aryna Sabalenka hit 24 winners to eliminate Britain's Katie Boulter 6-3, 6-3.
The second seed from Belarus advanced to a quarter-final against American Amanda Anisimova.
