Former world number one and 2003 champion Williams gave glimpses of why bookmakers have installed her as favourite for the women's singles with a 6-3, 6-0 dismissal of Madagascar's Dally Randriantefy.

As is often the case, Williams showed little consistency, dictating the proceedings with 18 unforced errors against nine from her opponent, but hitting 38 winners against just three from Randriantefy.

"It's definitely getting better and better with each match," Williams said. "I'm feeling really good physically. I'm really proud of how I'm doing, how I'm feeling and I'm really excited that I'm feeling so well."

Williams now plays teenager Sania Mirza on Friday, who made history by becoming the first Indian woman to reach the third round of a Grand Slam.

The 18-year-old from Hyderabad, ranked 166th in the world, confounded the form book to defeat Hungary's Petra Mandula 6-2, 6-1 in less than an hour.

Mirza, playing in her first Grand Slam event, can't wait for her date with Williams. "I'm really looking forward to finding out how hard she hits the ball," she said.

But while Williams and Mirza were in cruise control, it was a different story for Wimbledon champion Sharapova and world number two Mauresmo.

Siberian-born Sharapova looked in danger of wilting in the hothouse heat of her match against American journeywoman Lindsay Lee-Waters.

In the end Sharapova's class told and she prevailed 4-6, 6-0, 6-3.

The 17-year-old prodigy couldn't hide her relief after securing the win, which sets up a third-round meeting with China's Li Na on Friday.

"It was an amazing match. Both of us gave it all we've got, in the third set we just fought for every single ball," Sharapova said. "In the end it was just a matter of a few points."

Mauresmo had earlier struggled against Russia's Dinara Safina, going a set down before finally winning 2-6, 6-1, 6-0.

Elsewhere, Russian star Svetlana Kuznetsova briefly banished her Australian Open drugs nightmare with a dream performance to reach the third round.

The fifth-seeded US Open champion was never tested by Marion Bartoli of France and wrapped up a 6-2, 6-0 victory in just 40 minutes at Melbourne Park.

Kuznetsova has been caught in a drugs storm this week after authorities in Belgium revealed she had tested positive for the stimulant ephedrine at an exhibition event last month.

The 19-year-old fiercely denied being a dope cheat and took refuge from the controversy by bludgeoning her opponent with a near flawless performance.

"I'm a professional and I want to show that it doesn't affect me ... I didn't do anything wrong," said Kuznetsova.

But other seeds were not so lucky. Japan's 28th seed Shinobu Asagoe lost to China's Li, while Russian ninth seed Vera Zvonareva crashed 6-3, 6-3 to compatriot Vera Douchevina.

World number 64 Li, who made history last year by becoming the first Chinese player to win a title on the WTA Tour when she won in Guangzhou, advanced to the third round after downing Asagoe 6-3, 6-4 in 1hr 7min.

It was the latest in a series of notable results by Chinese women, who have come from nowhere in the past decade to establish themselves on the international circuit.