Kvitova, seeking to become the third Czech winner of the event in the last four years, defeated Vikhlyantseva 6-3 6-1 to set up a last-eight meeting with compatriot Katerina Siniakova.
Seeded eighth, Siniakova needed three sets to get the better of Ekaterina Alexandrova, ultimately winning 6-4 3-6 6-3.
Shuai Zhang and Mihaela Buzarnescu, seeded sixth and seventh, advanced to quarter-finals against Jasmine Paolini and Kristyna Pliskova respectively.
Australian veteran Sam Stosur, meanwhile, will face Camila Giorgi in the other last-eight match.
Mertens, meanwhile, had trailed early in her match against Laura Siegemund, but benefited from the retirement of the German due to injury to reach the quarter-finals of the Grand Prix de SAR La Princesse Lalla Meryem in Morocco.
The Belgian’s next opponent will be the experienced Sara Errani, who dispatched Johanna Larsson 6-3 6-0.
They were joined in progressing by Katarina Zavatska, Su-Wei Hsieh and Paula Badosa Gibert, who all needed three sets, the latter upsetting Polona Hercog 6-7 (5-7) 6-1 6-2 despite being ranked 150 places below the beaten Istanbul Cup finalist.
Ajla Tomljanovic, Aleksandra Krunic and Jana Fett won in straight sets.
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