Williams will now meet Italian Sara Errani for a quarter-final berth following the seventh seed's 6-4 6-4 win over Slovakian Daniela Hantuchova.
For the second successive day Williams needed three sets but she was glad with the way she recovered from a rusty opening.
"I made so many errors. She's one of those players that puts a lot of balls in the court, makes you play a lot of shots and I wasn't getting them in the court. It was good to turn it around," said Williams.
Pole Radwanska, playing in her first premier event as a top seed, said she was pleased with how she had settled in to the event.
"The first match is always the most difficult. Every week we play on different surfaces with different balls and in different conditions. I'm just happy I went through, especially in two sets, because of the heat," she said.
Radwanska will next face Sloane Stephens after the American beat fellow wildcard entrant Camila Giorgi of Italy 6-2 6-1 in an hour.
Stephens acknowledged she would be the underdog against the Pole.
"I'll just go out and play my game," Stephens said. "I'm not going to tell you how I'm going to play, because she'll probably read this and know how to play."
Radwanska's sister Urszula, now ranked 46th in the world, progressed to the third round after Kazakh Yaroslava Shvedova retired due to "heat illness".
Fourth seed Petra Kvitova made it through despite losing the first set to Mona Barthel. Germany’s Barthel broke and held to take the first but the Czech fought back to win 3-6 6-2 7-5.
One seed did go out early though; Slovakia’s Dominika Cibulkova withdrew injured after losing a set and a break to Andrea Hlavackova of the Czech Republic.
Denmark's Caroline Wozniacki needed just 59 minutes to dismiss Kazakh Sesil Karatantcheva 6-1 6-0 in a totally one-sided affair.
In the third round Wozniacki will face Russian Anastasia Pavlyuchenkova, who swept aside Germany's Julia Goerges 6-3 6-0.
And Ninth seed Li Na safely progressed to the next round, despite being taken to three sets by Romania's Sorana Cirstea. The former French Open champion won 6-3 4-6 6-3 in the late-finishing match on Center Court.