Russian Zvonareva coasted through the opening set against wildcard Nigina Abduraimova but was given more of a run for her money in the second set.
However she came through it to win 6-1 7-5 and set up a meeting with Slovak Kristina Kucova.
Second seed Anastasia Pavlyuchenkova defeated Russian compatriot Valeria Solovieva 7-5 6-4.
Pavlyuchenkova will next face Elena Bovina, who needed three sets to beat Zarina Diyas.
Third seed Elena Vesnina was forced to work hard for her place in the next round, battling past Yana Buchina 6-2 3-6 7-5.
The Russian looked to be firmly in control after easing through the first set before her compatriot bounced back to take the second and race out to a 2-0 lead in the third.
Vesnina called a medical timeout and received treatment to her right shoulder, before breaking back immediately on the resumption of play.
The 24-year-old held two match points on Buchina's serve but wrapped up the victory on her third attempt.
Next up for Vesnina will be Frenchwoman Aravane Rezai after she crushed 15-year-old local Kamilla Farhad 6-0 6-2.
Seventh-seeded Russian Ksenia Pervak also had a tough clash, being forced to recover from a set down before seeing off Thailand's Noppawan Lertcheewakarn 3-6 6-1 7-5.
Fifth-seeded Romanian Monica Niculescu was handed a simple passage into the second round when her opponent, Georgia's Eirini Georgatou, retired injured whilst trailing 5-1 in the first set.
Georgatou's compatriot Anna Tatishvili enjoyed a better day as she dismissed eighth seed Evgeniya Rodina 6-4 6-2 to set up clash with fellow Georgian Tatia Mikadze, who routed Russian Nina Khrisanova 6-0 6-0.
Ukraine's Kateryna Bondarenko and Belarusian Anastasiya Yakimova both progressed with straight sets wins over Lesia Tsurenko and Valeria Savinykh respectively.