“She started out really well, really played aggressive and I was a little too passive,” said Williams, who is seeking a 19th grand slam title to edge closer to record holder Steffi Graf, who won 22.

“Once I got down I thought ’You have done so well here, you have nothing to lose, have fun’ and I just started to play a lot better. Things really clicked. I had no other option but for things to click. I just had to start playing better.

“That’s one thing about the grand slams. You have to be ready for anyone at any stage. Playing Vera I had to get my mind like, ’Serena, this girl has been ranked as high as number two in the world’, she’s been on the tour and she’s a very quality player, she knows what to do. She knows how to win. I had to kind of snap into that.”

Zvonareva is currently ranked 203rd in the world but is competing in Melbourne on a special ranking after an injury-plagued 2014 which saw her play just five tournaments.

The 30-year-old Russian, who lost to Williams in the 2010 Wimbledon final, looked certain to take the opening set when she led 5-3 and forced three break points on the American’s serve, but Williams saved all three and finally won the game with an ace.

Williams took advantage of her reprieve to break back and powered through the rest of the contest to join elder sister Venus in the last 32.

Venus, who has not gone beyond the third round of any slam since Wimbledon in 2011, defeated compatriot Lauren Davis for the second tournament running.

“I’m just doing the best I can,” the 34-year-old said. “I always was, even when it wasn’t what I wanted. So whatever that is, I’m doing absolutely the best I can. I think as long as I’m doing my best, something good will come out of it.

“There is a scripture that says faith without works is dead. So you have to have faith, but you have work too. So I’m doing both.”

The Williams sisters withdrew from the doubles in Melbourne, apparently simply to concentrate on the singles, although Venus was rather coy on the subject.

“According to the rules you don’t have to give a reason. I think we’ll stick with that,” she said.

Pressed on whether it was due to the heat in Melbourne, she added: “No further questions on that. I object. Sustained. Thank you.”

Venus will play Italy’s Camila Giorgi in the third round and is scheduled to meet sixth seed Agnieszka Radwanska in the last 16. Radwanska appears to be enjoying life with new coach Martina Navratilova as she thrashed Sweden’s Johanna Larsson 6-0 6-1 in just 44 minutes.

Fourth seed Petra Kvitova was also a comfortable winner, the Wimbledon champion beating Germany’s Mona Barthel 6-2 6-4 to set up a third-round clash with 19-year-old American Madison Keys, who defeated 29th seed Casey Dellacqua.