Czech Kvitova struggled with her serve to lose the first set 6-3, committing five double faults and winning only 48 per cent of her first-serve points against the 65 per cent of the Spaniard.
And things only got worse in the second set as she was broken three times - replying with one break of her own - to lose it 6-2 as Suarez Navarro set up a third-round encounter with either Jelena Jankovic or Monica Niculescu.
Eleventh seed Ana Ivanovic came from a set down to beat American Varvara Lepchenko 4-6 7-5 6-1, the Serb converting all three of her break points in the deciding set to reach the last 16.
Fresh from a semi-final run in Seoul, world number 20 Lepchenko broke twice to win the opening set but lost momentum and went down in 129 minutes.
Russia's Elena Vesnina beat Austrian Tamira Paszek 7-6(5) 6-2 and local favourite Peng Shuai also reached the last 16 although it was well past midnight before she beat Yaroslava Shvedova 5-7 7-5 6-2.
In first round action, top seed Victoria Azarenka of Belarus eased past Frenchwoman Alize Cornet 6-1 6-0. She will play Sabine Lisicki after the German beat Anabel Medina Garrigues of Spain 6-3 5-7 6-2.
Fellow seeds Maria Sharapova and Caroline Wozniacki beat Simona Halep and Chanelle Scheepers 7-5 7-5 and 7-5 6-7 6-2 respectively.
Polona Hercog earned a second-round meeting with Ekaterina Makarova when Anastasia Pavlyuchenkova retired at 5-7 6-4 3-0 down while Zheng Jie was shocked 6-2 6-4 by Lara Arruabarrena Vecino and Sorana Cirstea defeated Sofia Arvidsson 6-2 6-3.