The men's No. 1 seed, 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.
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.
Williams said Zvonareva started out really aggressively "and I was a little too passive."
That's when she told herself: " 'You know Serena, you've done so well here, you have nothing to lose' — and I started playing a little better."
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.
Venus Williams earned 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.
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.
No. 24 Garbine Muguruza beat veteran Daniela Hantuchova, 6-1, 1-6, 6-0.
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.
Former USC star Steve Johnson beat No. 30 Santiago Giraldo, 6-3, 6-4, 6-2.