Williams converted 5 of 7 break-point chances against the unseeded Radwanska.
"My opponent started so strong," Williams said. "She went 4-1, but I was hoping to still win that set. After that, things went almost perfectly for me, so I'm really happy with that and to be playing again tomorrow."
Williams was a surprise first-round loser at the Australian Open last month, but appears to have regained the form that allowed the seven-time major winner to return to the top 10 late last year.
Williams will be playing Anastasija Sevastova of Latvia for the first time in the quarterfinals.
"I'm happy that I won, but it probably wasn't the nicest match we've both played," Voegele said. "It was difficult with the strong winds."