Strycova, who reached the Wimbledon semi-finals last month, battled for nearly two hours before losing to Pera, who is ranked 37 rungs below her in the WTA rankings.
"I was more focused on my game and finding that balance that is so important for me to play well," the 24-year-old said. "It's finding the balance between being too aggressive and being too passive.
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I think I was just able to find my balance and find the rhythm with my feet in the second set and keep going in the same way in the third.
Pera saved nine of 12 break points and converted six of her seven opportunities. The American saved four break points in her first two service games alone.
Strycova's poor service game cost her, with a double fault in the first game of the final set opening the door for Pera, and the Czech will have to improve quickly if she is to make an impact at the U.S. Open, which starts next week.
Pera will next face another Czech player in fifth seed Katerina Siniakova, who advanced after her Russian opponent Anastasia Potapova dropped the opening set 6-0 and retired midway through the second set 3-0 down.
China's Zhu Lin also retired on Tuesday with France's Alize Cornet moving into the last eight.
Cornet will face Italy's Camila Giorgi, who beat Germany's Andrea Petkovic 3-6 7-5 7-6(3) in a match that lasted two hours and 41 minutes. (Reporting by Rohith Nair in Bengaluru; Editing by Peter Rutherford )
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