Azarenka, who struggled in her first-round match, rolled over Dinah Pfizenmaier 6-1, 6-1. She had just nine winners but also just 11 unforced errors. Pfizenmaier matched the nine winners but hampered herself with 29 unforced errors.
Azarenka won three-quarters of the points on serve, saving both break points she faced.
Radwanska, seeded third, took out Williams 6-2, 6-3, closing the match with her fifth service break. Radwanska had just six unforced errors Wednesday while Williams had 33. Radwanska won half the points when Williams served.
That result came about 24 hours after Venus' sister Serena was upset in the first round for the first time in 47 Grand Slam events.
Eighth-seeded Marion Bartoli joined the Williams sisters on the sidelines after a 6-2, 3-6, 6-3 loss to Petra Martic. Bartoli had 14 double faults.
No. 6-seeded Samantha Stosur easily got past Irina Falconi 6-1, 6-4 without dropping serve, and 15th-seeded Dominika Cibulkova sailed past Vania King 6-0, 6-2, taking more than two-thirds of the points.
No. 10-seeded Angelique Kerber was down an early break 1-2 against Olga Govortsova in Wednesday's final women's singles match when play was suspended by rain.
Other seeded players winning Wednesday included 13th-seeded Ana Ivanovic, No. 18 Flavia Pennetta, 21st-seeded Sara Errani, No. 26 Svetlana Kuznetsova, No. 27 Nadia Petrova and 29th-seeded Anabel Medina Garrigues.
Also advancing to the third round were Sloane Stephens and Maria Jose Martinez Sanchez.