The 102nd-ranked Kleybanova rallied from 5-3 down in the second set to beat her Czech opponent. It was the Russian's first victory against a top-10 player since winning her battle against Hodgkin's lymphoma, which kept her out between May 2011 and March 2012.
The sixth-ranked Kvitova had been given a bye to the second round and was playing her first match since inspiring the Czech Republic to the Fed Cup final last weekend.
Earlier, Ana Ivanovic defeated Sabine Lisicki 6-1, 6-3 to send the Wimbledon finalist out in the first round. Ivanovic, the world No. 12 and former French Open winner, needed just one hour to beat the 24-year-old Berliner.
Italy's Flavia Pennetta fired 10 aces to none in her 7-5, 3-6, 6-1 win over wild card Andrea Petkovic. It left four of the six home players out of the most important German women's competition at the first hurdle.
Pennetta will face the fifth-seeded Jelena Jankovic in the second round, with the winner then playing Kleybanova in the quarterfinals.
Wild card Julia Goerges gave the supporters reason for cheer with a surprise 6-1, 7-5 win over Romania's Sorana Cirstea. The 94th-ranked German will next face Ivanovic on Thursday.