Both scored straight sets wins, with Radwanska beating Romanian Irina-Camelia Begu and Kerber successful over Czech Lucie Hradecka.
Australia's Sam Stosur will be in action later today against China's Jie Zheng.
German Kerber disposed of Hradecka in straight sets, completing a 6-3 6-1 victory.
Kerber has had her best year by far on tour, winning titles in Paris and Copenhagen and reaching the Wimbledon semi-finals and the French Open quarter-finals.
She will play the winner of the clash between American Madison Keys and 30th-seeded Austrian Tamira Paszek for a fourth-round berth.
Radwanska, who is unbeaten this year after tournament wins in Auckland and Sydney, eventually won 6-3 6-3 in 86 minutes.
Begu surprised the Polish player by breaking serve in the first game then fighting her way back into the match after Radwanska was serving for the first set at 5-1.
The Romanian also produced a comeback in the second set as Radwanska let go of a number of chances.
"I'm very happy to be playing my best tennis at the the start of the year," Radwanska said.
"It's always great to win in two sets."