The fourth-seeded Williams -- looking for her first New York title since 2008 --- rolled to a 6-4, 6-0 win over Russia's Ekaterina Makarova, gaining some revenge after Makarova beat her in the fourth round of the Australian Open.
The American star's next opponent will be the Czech Republic's Andrea Hlavackova, who staged a come-from-behind 5-7, 6-4, 6-4 decision over 14th-seeded Russian Maria Kirilenko.
Poland's Radwanska, meanwhile, bested Jelena Jankovic of Serbia 6-3, 7-5. Next up for her will be Roberta Vinci of Italy, who notched a 6-2, 7-5 win over 13th-seeded Slovak Dominika Cibulkova.
Also Saturday, No. 6 Angelique Kerber of Germany pounded Belarusian Olga Govortsova, 6-1, 6-2, advancing to face 10th-seeded Italian Sara Errani, who cruised past Russian Olga Puchkova 6-1, 6-1 Saturday.
Serbia's Ana Ivanovic, seeded 12th, also reached the Round of 16, besting American Sloane Stephens.
Her next opponent will be Bulgaria's Tsvetana Pironkova, who outlasted Spaniard Silvia Soler-Espinosa 6-1, 6-7 (3-7), 6-3.