Williams’ 6-4 6-4 success sets up a 12th career meeting with the 29th seed, who had to come from a set down to overcome Spaniard Virginie Ruano Pascual 4-6 6-1 6-1.
Williams is relishing the opportunity to gain her revenge for a last-16 defeat at the US Open last year.
She said: “I haven’t played her in a while so I’m looking forward to it. I had no idea I might play her and I’m excited about that.
“She’s been struggling a lot this year so you can only hope she does well, especially being a past champion here and a past champion in general. It will be a good match.”
Williams was in far from scintillating form against the 17-year-old Radwanska, hitting 22 unforced errors but still having too much for her opponent in the course of her one hour and 20 minute success.
Williams revealed afterwards she had felt the pressure during the match, while the prospect of playing on Court Two — the so-called ‘Graveyard of Champions’ — had also been playing on her mind.
“I felt I played okay but I was a little tight out there,” she continued.
“I just didn’t want to go out on Court Two and I was happy to get the first set under my belt. It was windy too and the conditions weren’t great. I felt like I should just get it over with.”
Fourth seed Svetlana Kuznetsova eased past Ukrainian Kateryna Bondarenko 6-2 6-3 on Centre Court to book her place in the last 32.
Kuznetsova, who will celebrate her 23rd birthday on Friday, showed none of the frailties which forced her to edge past French qualifier Mathilde Johansson in three sets.
Kuznetsova double-broke immediately to take a 3-0 lead and from that moment the outcome was never in doubt as she coasted home with a solid grasscourt performance.
Eighth seed Anna Chakvetadze enjoyed a similarly straightforward passage as she overcame Romania’s Edina Gallovits 6-4 6-2 on Court Two.
Fifth seed Elena Dementieva dropped a set before disposing of Maria Elena Camerin 6-3 6-7 (7/9) 6-3 in a match suspended late last night at 3-3 in the second set, while 18th seed Nicole Vaidisova survived a major blip to beat Australian Sam Stosur 6-2 0-6 6-4.
Last year’s finalist Marion Bartoli and 16-year-old Russian Anastasia Pavlyuchenkova also went through, but 20th seed Francesca Schiavone fell 3-6 7-5 9-7 to Anabel Medina Garrigues of Spain.