Five-time champion Williams moved on with her 600th career win, breezing past 42-year-old Kimiko Date-Krumm of Japan 6-2, 6-0, unleashing eight aces and stretching her career-best winning streak to 34 matches.
Williams' next opponent will be 23rd-seeded German Sabine Lisicki, who advanced Saturday with a 4-6, 6-2, 6-2 win over Australia's Samantha Stosur.
Poland's Radwanska, who was runner-up at Wimbledon last year, out-dueled American Madison Keys 7-5, 4-6, 6-3, and advanced to a fourth-round match against Tsvetana Pironkova of Bulgaria.
Pironkova advanced with a three-set victory over Petra Martic of Croatia.
Briton Laura Robson overcame a sluggish first set Saturday to beat New Zealand's Marina Erakovic 1-6, 7-5, 6-3. She'll next face Estonia's Kaia Kanepi, a logged a 6-2, 6-3 victory over American Alison Riske.
In matches begun Friday but halted due to darkness, No. 8 seed Petra Kvitova of the Czech Republic advanced with a 6-3, 2-6, 6-3 win over Russian Ekaterina Makarova, and 17th-seeded American Sloane Stephens battled her way to a 7-6 (7-3), 0-6, 6-4 victory over Czech Petra Cetkovska.
In other third-round matches completed Saturday, Spain's Carla Suarez and Puerto Rico's Monica Puig each recorded a victory.