German Kerber, who also suffered a surprise loss to Elina Svitolina in the round of 16 at last week's China Open, made six double faults and struggled on serve throughout as her 38th-ranked opponent secured victory in an hour and 14 minutes.
Moscow-born Gavrilova, 22, fired down five aces as she continues her quest for a first WTA singles title and will next face Frenchwoman Kristina Mladenovic for a place in the final.
Her victory over Kerber, who surged to the top of the rankings after winning the Australian and U.S. Open titles this year as well as reaching the Wimbledon final, was her first against the German Olympic silver medallist in six meetings.
"I guess I'm really excited, I've played Angie six times and now I've finally won, it's really exciting," Gavrilova said.
"I was prepared for a battle. I think Angie has had a long season. I didn't expect to win that easily but I was ready to turn it around and get a win against her."
Mladenovic defeated American Bethanie Mattek-Sands 3-6 6-3 7-6(3).
Unseeded Cornet, who ousted second seed Venus Williams in the previous round, committed 13 double faults as seventh seed Jankovic secured a 3-6 6-4 6-2 win to set up a semi-final clash with Denmark's Caroline Wozniacki.
Former world number one Wozniacki, the fifth seed, took just under 90 minutes to oust China's Wang Qiang 6-3 7-5.
(Reporting by Ian Rodricks in Bengaluru; Editing by John O'Brien and Ed Osmond)
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