Russia's Sharapova posted a straight-set 7-5, 6-3 victory over Carla Suarez Navarro in a 1-hour, 40-minute match at Indian Wells, gaining the round of 16.
There she'll meet Lara Arruabarrena-Vecino of Spain, who notched a 2-6, 6-4, 6-4 upset of 14th-seeded Italian Roberta Vinci.
Meanwhile, Poland's Radwanska had to battle through a 2 1/2-hour match with 27th-seeded Sorana Cirstea of Romania to emerge with a 6-7 (3-7), 6-3, 6-4 victory.
Her next opponent will be No. 13 seed Maria Kirilenko of Russia, who got past American Mallory Burdette 6-3, 4-6, 6-2.
In other third-round action Sunday, fifth-seeded Petra Kvitova of the Czech Republic bested Ukrainian qualifier Lesia Tsurenko 6-2, 7-6 (7-5); No. 6 seed Sara Errani of Italy dispatched Sweden's Johanna Larsson 6-3, 6-1; and No. 19 Czech Klara Zakopalova downed 12th-seeded Slovakian Dominika Cibulkova 6-4, 7-5.