World number one Kerber, who was leading 2-1 in the third set when rain halted play on Wednesday, was in no mood to hang around on resumption and quickly booked a last eight meeting with Australian Daria Gavrilova.
American Williams was beaten by unseeded Frenchwoman Alize Cornet 6-2 3-6 6-2.
Williams committed six double faults and struggled on serve in her defeat by Cornet, who next takes on defending champion Jelena Jankovic.
China's Wang Qiang received a walkover after Britain's Johanna Konta, who occupies the final qualifying berth in the race to qualify for the season-ending WTA Finals in Singapore, pulled out with an abdominal injury.
Denmark's Caroline Wozniacki, the fifth seed, beat Britain's Heather Watson 6-3 6-3 and American Bethanie Mattek-Sands overcame sixth seed Caroline Garcia of France 4-6 6-2 6-4.
(Reporting by Ian Rodricks in Bengaluru, editing by Pritha Sarkar)
