Kuznetsova, Sharapova move on at Pacific Life Open
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World No. 3 Kuznetsova, who lost to another Slovakian, Daniela Hantuchova, in last year's final, came back from an error-prone second set to seal victory in one hour, 48 minutes.
Kuznetsova was relieved after booking her place in the fourth round of the women's draw at the Indian Wells Tennis Garden.
"I just stopped a little bit and lost my concentration," the Russian said of her slump midway through the match.
"I have to figure out how to start playing better earlier after being 5-0 down in the second set. I didn't want to go to the third set 6-0 down so I changed that to give me the key to play better in the third."
Kuznetsova, initially dominating the baseline exchanges, cruised through the opening set after breaking her opponent in the second and sixth games.
However, the Russian inexplicably began to over-hit her groundstrokes and was broken in the second and fourth games of the second set to trail 0-5.
She mounted a fightback to close to 4-5 but Cibulkova again broke in the 10th game to level the match.
Kuznetsova, a runner-up this year in Sydney and Dubai, upped her game in the third set to book a place in the next round against Denmark's Caroline Wozniacki.
The 17-year-old Dane, who won the 2006 Wimbledon junior title, brushed aside Aiko Nakamura of Japan 6-0, 6-2 in 53 minutes.
Meanwhile, Hantuchova, who is seeded fifth, beat China's Jie Zheng in straight sets, American Ashley Harkleroad upset No. 8 Dinara Safina, No. 15 Alona Bondarenko topped No. 17 Amelie Mauresmo and No. 10 Agnieska Radwanska got by in three sets against Peng Shuai.