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The former finalist won the opening set 6-4 when the match got underway yesterday, but the British number five dug deep to level in a tie-break.
As the match resumed today, it looked like Moore would have the edge as she picked up a break early in the deciding set.
However, the Russian, who returned to the women's tour after an 18-month break earlier this year, fought back to get the set back on serve at 5-5.
Zvonareva edged ahead to take an 8-7 lead before a couple of errors from her 21-year-old allowed her to seal the match 6-4 7-6(3) 9-7.
Zvonareva will play Croatia's Donna Vekic in the next round.