American world number 36 Vandeweghe then took five games in a row to turn the match on its head and stood toe to toe with the Wimbledon champion in the decider before faltering.
Kvitova, also taken to three set by Olga Govortsova in the first round, broke in the eighth game when her opponent netted a forehand and she completed victory with a flourish.
There were no alarms for defending champion Maria Sharapova as she crushed Colombia's Mariana Duque-Mariño 6-1 6-2.
Former runner-up Victoria Azarenka set up a third-round clash with top seed Serena Williams with a 6-3 6-3 defeat of Croatian Ajla Tomljanovic.
Topping the early men's programme on Tuesday was Bulgaria's Grigor Dimitrov and French showman Gael Monfils, both of whom enjoyed easy victories on the red clay.
Tenth seed Dimitrov was 6-4 3-0 ahead when American opponent Donald Young quit with an injury while Monfils dropped only two games against Serbian Viktor Troicki.
Top seed Roger Federer was not in action.
(Reporting by Martyn Herman, editing by Pritha Sarkar)
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