Serena Williams, seeking a sixth Australian Open and 19th Grand Slam title, saved three set points in the ninth game before holding to swing the momentum of the match.
She said Zvonareva, a two-time Grand Slam finalist who played only five tournaments in 2014, started out aggressive "and I was a little too passive."
She said she told herself: "You know Serena, you've done so well here, you have nothing to lose."
"And I started playing a little better," she said.
After breaking Williams' serve twice in the first set, Zvonareva won only eight points in the second set and didn't get close to another break-point chance.
Venus Williams had a 6-2, 6-3 win against 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 in 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.
The Associated Press contributed to this report.
Serena Survives First-Set Scare
Serena Williams saved three set points in the ninth game before holding to win the set and that match, beating Vera Zvonareva 7-5, 6-0 in the second round at the Australian Open.
Venus Rolls Davis
Venus Williams topped Lauren Davis 6-2, 6-3 to advance to the third round at the Australian Open.