Her last two trips to the Australian Open have ended in quarterfinal losses to the eventual champion Kim Clijsters in 2011 and Victoria Azarenka last year.
With top-ranked Azarenka and Williams on the other half of the draw, Radwanska's main obstacles to reaching the final are 2011 French Open champion Li Na, who beat Olga Goortsova of Belarus 6-2, 7-5 to reach the third round, and potential semifinal rivals Maria Sharapova or No. 5 Angelique Kerber of Germany.
Kerber advanced Wednesday with a 6-3, 6-1 win over Lucie Hradecka and No. 2-ranked Sharapova, the reigning French Open champion, was scheduled to play later Wednesday against Japan's Misaki Doi.
Williams hurt her ankle in her opening victory Tuesday, causing speculation that her bid for a third consecutive major title Open could be in jeorpardy. She has only lost one match since her first-round exit at the French Open last year.
In early women's second-round matches Wednesday, No. 18 Julia Gorges beat Romina Oprandi 6-3, 6-2; Russian qualifier Valeria Savinykh upset No. 15 Dominika Cibulkova 7-6 (6), 6-4; and Kirsten Flipkens of Belgium ousted No. 23 Klara Zakopalova 6-1, 6-0.
On the men's side, 10th-seeded Nicolas Almagro was first through to the third round with a 6-4, 6-1, 6-2 win over fellow Spaniard Daniel Gimeno-Traver; No. 16-ranked Kei Nishikori of Japan beat Carlos Berlocq 7-6 (4), 6-4, 6-1; and No. 20 Sam Querrey advanced when fellow American Brian Baker retired in the second set with what appeared to be a right knee injury. Baker, who returned to the tour last year after almost six seasons out with a variety of injuries, won a first-set tiebreaker and was even at 1-1 on serve in the second. He was later pushed from Court 6 in a wheelchair.
"Yeah, he's the last person that deserves anything like that with his five or six surgeries already," Querrey said. "He does everything right, treats his body great, just trying to come back, and then something like that happens, it's just so unlucky."
Williams had a moment on day two when she felt like her tournament could be over.
Flat on her back, her sore right ankle raised and her hands covering her face, she tried to block out thoughts that her bid for a third straight Grand Slam title might be ruined.
"I almost panicked, and I thought, 'I can't do that,'" she said. "I just have to really remain calm and think things through."
The stats showed this was nothing more than a stroll a 6-0, 6-0 wipeout in 54 minutes of No. 110-ranked Edina Gallovits-Hall at Melbourne Park.
But the match took significantly longer to complete give a medical timeouts and other treatment. And while the score may have been painful to her opponent, there was plenty of pain to go around.
The first set was 4-0 after 19 minutes at Hisense Arena when her tumble near the baseline diverted attention from center court, where a day session featuring Roger Federer, Andy Murray and women's champion Azarenka was under way.
After some deep breaths, the 31-year-old Willlams pulled herself together, got to her hands and knees for a few minutes and gradually to her feet.
Her already heavily taped ankle was assessed and re-taped. She went back on court and won the next four points to get herself to another changeover, and more attention from the doctor. She went back and held another service game to clinch the set, giving her time for more treatment.
"A very similar thing happened to me last year, almost on the same side, the same shot," Williams said, referring to a fall that forced her to pull out of the Brisbane International last year and contributed to her fourth-round exit at the Australian Open.
Willilams said she expects to at least start her second-round match Thursday against Spain's Garbine Muguruza, who beat Magdalena Rybarikova of Slovakia 4-6, 6-1, 14-12.
"Oh, I'll be out there," Williams said. "I mean, unless something fatal happens to me, there's no way I'm not going to be competing. I'm alive. My heart's beating. I'll be fine."
She's still hopeful that if results go according to rankings, she'll meet Azarenka in the semifinals. The 23-year-old Belarusian had a 6-1, 6-4 opening win over Monica Niculescu.
She progressed along with former No. 1 Caroline Wozniacki, who beat Sabine Lisicki of Germany 2-6, 6-3, 6-3, and American teenager Sloane Stephens, who beat Simona Halep of Romania 6-1, 6-1.
In a matchup of two major winners, 2011 Wimbledon champion Petra Kvitova defeated 2010 French Open titlist Francesca Schiavone 6-4, 2-6, 6-2.
And 42-year-old Kimiko Date-Krumm upset 12th-seeded Nadia Petrova of Russia 6-2, 6-0 to set a record for being the oldest woman to win a singles match at the Australian Open.
"Some players' mothers are younger than me," she said, laughing.
Murray, playing his first match at a major since winning the U.S. Open and breaking a 76-year drought for British men at Grand Slam tournaments, defeated Robin Haase of the Netherlands 6-3, 6-1, 6-3.
Second-ranked Federer beat Benoit Paire of France 6-2, 6-4, 6-1 in his first competitive match of the season. Other men progressing included 2009 U.S. Open champion Juan Martin del Potro, 2008 finalist Jo-Wilfried Tsonga and big-serving Canadian Milos Raonic.
R Agnieszka Radwanska's 6-3, 6-3 win keeps her undefeated this year.
• For the men, American Sam Querrey advances after Brian Baker retires with a knee injury.