Serena Williams beat Kiki Bertens of the Netherlands 7-6 (7-5) 7-5 in round two and will play her sister Venus in the third round at Indian Wells.
Defending champion Elena Vesnina of Russia also moved into the third round after she beat American Catherine Bellis 2-6 6-1 6-1.
Elina Svitolina defeated German Mona Barthel 6-4 6-3.
Caroline Wozniacki beat Spaniard Lara Arruabarrena 6-4 6-1.
Grigor Dimitrov of Bulgaria lost 7-6 (7-4) 4-6 6-3 to Spain’s Fernando Verdasco.
World number one Roger Federer took the first set against Argentina’s Federico Delbonis 6-3. Second set was level at 2-2 when play was suspended for the day because of rain and match will resume on next day.
Austrian fifth seed Dominic Thiem beat Stefanos Tsitsipas of Greece 6-2 3-6 6-3.