World number 16 Vandeweghe will take on the winner of the match between American Alison Riske or Hungarian qualifier Fanny Stollar in the last eight.
Russian Veronika Kudermetova, who stunned defending champion Anett Kontaveit in the first round, reached her first WTA quarter-final after ousting Switzerland's Belinda Bencic 6-2 2-6 6-3.
The 21-year-old will face Serbian Aleksandra Krunic in the last eight after the seventh seed eased to a 7-5 6-2 win over Bibiane Schoofs of the Netherlands.
(Reporting by Aditi Prakash in Bengaluru, editing by Ed Osmond)
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