Sharapova got past Hsieh Su-Wei 6-1, 6-4 in windy conditions. Sharapova had five double faults, including two in a row that gave Hsieh a break and a 2-3 lead in the second set.
After Hsieh held serve to go up 4-2, Sharapova charged back and won four consecutive games and the match.
Radwanska won the first nine games in a 6-0, 6-2 rout of local-favorite Heather Watson. Radwanska took nearly 75 percent of the points in the first set and faced two break points on the day.
She claimed 60 percent of the points off Watson's serve.
Eighth-seeded Angelique Kerber lost serve once in a 6-2, 6-3 victory over 28th-seeded Christina McHale. Kerber fought off 10 break points, while converting both of her own, in taking the first set.
She draws Kim Clijsters in the fourth round. Clijsters had built a 6-3, 4-3 lead when 12th-seeded Vera Zvonareva retired with a respiratory illness.
No. 15-seeded Sabine Lisicki defeated Sloane Stephens 7-6 (7-5), 1-6, 6-2. Lisicki, who had to overcome 35 unforced errors, lost her serve just once outside of the second set. The win put her into the fourth round against Sharapova.
Also, 30th-seeded Peng Shuai was on the court just 47 minutes in a 6-1, 6-2 romp over Arantxa Rus.
Late Friday 17th-seeded Maria Kirilenko defeated Sorana Cirstea 6-3, 6-1 and Camila Giorgi upset No. 20 Nadia Petrova 6-3, 7-6 (8-6). Kirilenko draws Peng in the next round and Giorgi plays Radwanska.
In a second-round result Friday, Jana Cepelova knocked off 26th-seeded Anabel Medina Garrigues 6-7 (7-9), 7-6 (7-5), 6-3. The total time of the match, which began Thursday, was 3 hours, 6 minutes. Cepelova held off six break points in the third set.
Disney's 'Jessie' to feature network's first engagement
Rosie O'Donnell unveils nearly 50-pound weight loss