Azarenka needed just 55 minutes to run over Eleni Daniilidou 6-1, 6-0. She won more than two-thirds of the points in the match, including 74 percent when serving. She had 21 winners and 13 unforced errors.
Williams won 78 percent of the points on serve in a 6-2, 6-0 win against Garbine Muguruza. Williams pounded out 10 aces and saved the only break point she faced while breaking Muguruza's serve five times.
Laura Robson wrapped up the second-round with a 2-6, 6-3, 11-9 upset of eighth-seeded Petra Kvitova. Both players had late breaks, including one by Kvitova that made the score 6-6, to force extra games. But Robson recorded her fourth break of the set for a 10-9 lead and held at love to finish off the win. The final set took 94 minutes.
Kvitova had 18 aces but also 18 double faults and 42 winners against 50 unforced errors.
Also Thursday, Caroline Wozniacki, the No. 10 seed, defeated Donna Vekic 6-1, 6-4; 14th-seeded Maria Kirilenko advanced 7-5, 6-2 over Peng Shuai; and No. 16 Roberta Vinci stopped Akgul Amanmuradova 6-3, 6-2.
There were three upsets Thursday. Bojana Jovanovski took a tight 7-5, 7-5 decision over No. 17-seeded Lucie Safarova; Elena Vesnina stopped No. 21-seeded Varvara Lepchenko 6-4, 6-2; and Svetlana Kuznetsova knocked off 26th-seeded Hsieh Su-Wei 6-2, 6-1.
Among the seeded players advancing to the third round were No. 20 Yanina Wickmayer and No. 29 Sloane Stephens while unseeded players, in addition to those with upset wins, winning Thursday were Ayumi Morita, Jamie Hampton, Kimiko Date-Krumm, Carla Suarez Navarro and Lesia Tsurenko.
Azarenka will play Hampton in the third round and Williams next faces Morita.
Notable deaths of 2014 [PHOTOS]