Williams led throughout a 6-2, 6-4 win over Azarenka. She is 3-0 versus Azarenka in her career, not having dropped a set.
Williams, whose play has been limited by health issues that have dropped her ranking to No. 63, capitalized on serving issues for Azarenka in posting three breaks in the first set. Williams had two more breaks in the second set, including one that put her in position to serve for the match at 5-4.
Azarenka was one of four upset victims Tuesday with No. 3-seeded Sara Errani, ninth-seeded Sloane Stephens and No. 14 Kirsten Flipkens all losing second-round matches.
Svetlana Kuznetsova improved to 6-1 all-time against Errani with a 6-4, 6-4 win that included five breaks. Eugenie Bouchard ousted Stephens 5-7, 7-6 (9-7), 6-3 with two third-set breaks. Magdalena Rybarikova stopped Flipkens 6-2, 6-3 without losing her serve.
Second-seeded Agnieszka Radwanska, coming off a tournament win Sunday in Seoul, advanced to the third round with a 7-6 (7-5), 6-0 decision over Aleksandra Wozniak, winning 12 of 14 points receiving serve in the second set.
Other seeded players advancing Tuesday were sixth-seeded Jelena Jankovic, No. 11- Ana Ivanovic, 12th-seeded Samantha Stosur, No. 13 Simona Halep, 15th-seeded Sorana Cirstea and No. 16 Dominika Cibulkova.
Madison Keys came out on top of a 7-5, 6-2 result over Peng Shuai in the only match Tuesday between unseeded players.
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