Garcia, who had beaten four-times Grand Slam winner Naomi Osaka in straight sets on Sunday, continued her inspired form as she defeated Gauff on her home court in one hour 43 minutes.
After a thrilling encounter, which ebbed and flowed, Garcia, ranked 27th in the world, emerged to face a last eight meeting with the winner of Monday's match between Romanian Sorana Cirstea and American Danielle Collins.
The victory was Garcia's first against a top ten opponent since late 2022.
Fourth seed Elena Rybakina defeated American Madison Keys 6-3 7-5 to set up a quarter-final with Greece's Maria Sakkari.
Sakkari advanced with a walkover after Anna Kalinskaya had to withdraw ahead of their match on Monday due to a right thigh injury.
Victoria Azarenka also progressed with a 7-5 6-1 win over Britain's Katie Boulter.
World number one Iga Swiatek is in action later on Monday against Ekaterina Alexandrova.