"Venus is going to have a tough match," Zvonareva said. "I will be prepared and hope to raise the level of my game."
Zvonareva was broken in the first game of the match, but she broke back in the eighth and then won the tiebreaker.
"I could have played better," the seventh-seeded Russian said. "I didn't serve too well and I was trying to hit the ball too hard. I am not happy with the way I constructed my points."
In the second set, Zvonareva jumped out to 5-0 lead, breaking Szatmari in the second and fourth games.
In January, Zvonareva rolled her left ankle during a practice session at the Hobart International. She then retired in the first round of the Australian Open, but returned last month and reached the final at the Qatar Open.
"Coming back from injury is tough," Zvonareva said. "I am slowly getting my self confidence back and hope to do well."
Zi Yan also advanced to the quarterfinals, beating Olga Savchuk of Ukraine 6-2, 6-1.
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