Barty’s breakthrough win also scuppered world No.2 Kvitova’s hopes of claiming the No.1 ranking for the first time.
The 29-year-old needed to lift the trophy to usurp Naomi Osaka, who lost in the third round.
Barty will face Anett Kontaveit in the semi-finals after the Estonian brought Hsieh Su-wei’s giant-killing run to a dramatic end with a thrilling comeback win over the Taiwanese.
The 33-year-old Hsieh, having taken out Osaka and 13th seed Caroline Wozniacki, looked poised to add the 21st seed to her list by racing to a 4-0 lead in the third set.
But Kontaveit, 10 years her junior, took a page from the Taiwanese player’s own playbook and dug in to prevail 3-6, 6-2, 7-5.