The No.2-seeded Williams came out firing, reeling off the first eight games of the match to build a 60 20 lead; Radwanska got on the board with a break in the next game and steadied herself against Williams but the big-hitting American managed to close the match out in straight sets anyway, 60 63.
"Every time I play her she improves," Williams said. "I didn't make a lot of errors in the first set, and then I started making some errors that were just ridiculous. She started playing a lot better.
"It was very windy, but I felt good out there. I came through."
Williams has now won 58 of her last 61 matches, a stretch dating back to the beginning of the clay court season last April - the only three losses were to Virginie Razzano (French Open first round), Angelique Kerber (Cincinnati quarterfinals) and Sloane Stephens (Australian Open quarterfinals).
Should Williams reach the semifinals at this week's Qatar Total Open she will secure a return to the No.1 ranking next week, which would make her the oldest player ever to be No.1 and give her a 124th career week at the top spot, a total only bettered by five of her fellow WTA legends: Steffi Graf (377), Martina Navratilova (332), Chris Evert (260), Martina Hingis (209) and Monica Seles (178).
"It's awesome," Williams said of the stat. "Since I didn't do the youngest, I figure I'll do the oldest."
Standing in between Williams and the No.1 ranking will be No.7 seed Petra Kvitova, who dropped the first set and rallied from behind in both the second and third sets - 0-3 in the second set, 1-3 in the third set - to edge No.11 seed Nadia Petrova in a two-hour, eight-minute marathon, 46 64 63.
The other quarterfinal match-up on the bottom half features No.3 seed Maria Sharapova and No.8 seed Samantha Stosur, after Sharapova outslugged Klara Zakopalova, 63 63, and Stosur ended the run of Svetlana Kuznetsova, 64 75. Kuznetsova had taken out No.9 seed Marion Bartoli the day before.
"The conditions are certainly tough," Sharapova said. "I think the windiest match I've played in my career was here many years ago, five years ago, I think it was the first round. It was pretty brutal.
"Today was one of those days where you go out there and almost know you might not be playing your best tennis. I certainly didn't play my best tennis today. But it's good to get through in these conditions. And Klara is a really tricky player to play against. I've definitely had trouble against her in the past, three-setter at Roland Garros last year. She can play - she hits the ball pretty big out there."
How does one get through such conditions successfully? "I always feel these kinds of conditions even out the level of tennis in a way that it doesn't really matter who you're playing," Sharapova added. "You might have to play a little bit safer and not as close to the lines. Sometimes any game plan you have goes out the window just because the circumstances of how you feel out on the court are so different to maybe even the warm-up. So 'adjustment' is a big word during these types of days."