Men's No. 1 Novak Djokovic had an easier run, spending a minute less than Williams on Rod Laver Arena in his 84-minute, 6-0, 6-1, 6-4 win over No. 88-ranked Andrey Kuznetsov on Thursday.
Canadians Vasek Pospisil and Milos Raonic advanced to the third round. Pospisil, from Vancouver, beat Italy's Paolo Lorenzi 6-7 (3), 7-6 (4), 6-3, 6-4 while Raonic, from Thornhill, Ont., defeated American Donald Young 6-4, 7-6 (3), 6-3.
Defending champion Stan Wawrinka, who ended four-time champion Djokovic's 25-match winning streak at Melbourne Park in the quarterfinals last year, advanced to the third round for the seventh consecutive year with a 7-6 (4), 7-6 (4), 6-3 win over No. 194-ranked qualifier Marius Copil.
Williams, seeking a sixth Australian and 19th Grand Slam title, saved three set points in the ninth game before holding to swing the momentum of her match against Zvonareva, a two-time Grand Slam finalist who played only five tournaments in 2014.
That's when she told herself: "'You know Serena, you've done so well here, you have nothing to lose.'"
"Things really clicked. I had no other option but for things to click," she added. "Yeah, I just had to start playing better."
Williams said she had to transition quickly to get past such an experienced player in the second round.
She said she was thinking: "This girl has been ranked as high as No. 2 in the world ... she's a very quality player, she knows what to do."
After breaking Williams' serve twice in the first set, Zvonareva only won eight points in the second set and didn't get close to another break-point chance.
Williams will next play No. 26 Elina Svitolina, who beat Nicole Gibbs 7-6 (3), 7-6 (6).
Venus Williams had a 6-2, 6-3 win against fellow American Lauren Davis. The elder of the Williams sisters, a seven-time major winner, hasn't been beyond the third round at a major since reaching the fourth round at Wimbledon on 2011.
Wimbledon champion Petra Kvitova, seeded No. 4, beat Mona Barthel 6-2, 6-4 to move into a third-round match against Madison Keys, who rallied to beat No. 29 Casey Dellacqua 2-6, 6-1, 6-1.
Sixth-seeded Agnieszka Radwanska needed only 44 minutes to race through her 6-0, 6-1 win over Johanna Larsson, continuing her run of reaching the third round or better at every Australian Open since 2010. In the next round, she'll face No. 30 Varvara Lepchenko, a 6-1, 7-6 (1) winner over Ajla Tomljanovic.
No. 11 Dominika Cibulkova, the 2014 finalist, No. 19 Alize Cornet and No. 24 Garbine Muguruza also advanced.
On the men's side, U.S. Open finalist Kei Nishikori beat Ivan Dodig 4-6, 7-5, 6-2, 7-6 (0), getting a crucial service break in the 10th game of the fourth set when Dodig was serving for the set and then dominating in the tiebreaker. He will next face Steve Johnson, who beat No. 30 Santiago Giraldo 6-3, 6-4, 6-2.
No. 12 Feliciano Lopez was leading 4-6, 4-6, 7-6 (3), 4-0 when Adrian Mannarino retired from their second-round match and No. 19 John Isner beat Andreas Haider-Maurer 6-4, 7-6 (4), 4-6, 6-4.