Azarenka, the defending tournament champion, stopped Romina Oprandi 6-2, 6-3 by taking 60 percent of the points in the match. Azarenka finished off four of her five break chances.
She had three breaks on the first set and only needed one to win the second because she dropped only five points in five games serving in the set.
Radwanska was a 6-3, 6-2 winner over Anastasia Rodionova. She won 71 percent of the points on serve, including taking all three of Rodionova's break opportunities.
No 6-seeded Sara Errani had an even more one-sided 6-4, 6-0 victory over Timea Babos. Errani lost just seven points in the second set.
Eighth-seeded Samantha Stosur was pushed to a third set before advancing with a 6-4, 1-6, 6-3 decision over Monica Niculescu and Caroline Wozniacki, the No. 10 seed, defeated Sorana Cirstea 7-6 (9-7), 6-0 while 11th-seeded Nadia Petrova got by Hsieh Su-Wei 3-6, 6-2, 6-4.