The 26th-seeded Spaniard came through 6-1, 6-2 for her best Grand Slam result since she considered quitting the sport due to a serious back injury.
Despite the one-sided scoreline, the first two games on Sunday took 17 minutes to complete while Badosa saved all eight break points she faced.
"It was so humid I thought I was going to die," said the 26-year-old New York-born player of the humidity which hit 85% at noon.
"I knew she was tough but I'm tougher."
Badosa is the first Spanish woman to make the quarter-finals since Carla Suarez Navarro in 2018 while it has been 28 years since a Spaniard last reached semi-finals.
That honour fell to Conchita Martinez in 1996.
Badosa will take on either defending champion Coco Gauff or Emma Navarro for a place in the semi-finals.
