The American converted five of seven break-point chances on Thursday against the unseeded Radwanska and set up a last-eight match-up with Anastasija Sevastova of Latvia.
Williams was a surprise first-round loser at the Australian Open last month, but appeared to have regained the form that allowed the seven-time major winner to return late last year to the world's top 10.
In other second-round matches, No.2 seed Misaki Doi of Japan scored a 6-2 6-3 win over Magda Linette of Poland and Stefanie Voegele of Switzerland upset Chinese fifth seed Zheng Saisai 6-4 6-4.
Source: WTA Tennis
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