The Russian had won their three previous meetings and looked on course to book her place in the next round when she took the first set 6-4, but Slovakian Cibulkova had other ideas and after taking the second set on a tie-break to level the match, managed to stay with Zvonareva and battle to a 4-6 7-6 6-4 victory.
The win ranks as one of the best wins of her career and is the biggest upset of the BNP Paribas Open so far.
Cibulkova now goes on to face 23rd seed Wickmayer of Belgium who came through at the expense of 14th seed Kaia Kanepi 3-6 6-4 6-1.
