The Australian Open finalist already has a WTA singles title to her name this year and is on course to add to her Acapulco success after a 6-1 6-3 victory over the Turkish qualifier.
Cibulkova was always likely to be too strong for the world number 216 and so it proved as she raced into a 5-0 lead before wrapping up the opening set in 27 minutes at the WTA International hard-court event.
The second was a trickier test for the Slovakian who found herself 2-1 and 3-2 down before reeling off the final four games to progress.
Fourth seed Patricia Mayr-Achleitner was made to work harder by 19-year-old Montenegrin Danka Kovinic but she eventually progressed 6-4 3-6 6-1 despite wasting two of three match points in the third set.
Italian qualifier Giulia Gatto-Monticone saw her run in the competition end at the second round as she crashed out 6-1 6-4 to number six seed Zarina Diyas.
Gatto-Monticone had seen off Tunisian Jabeur Ons in the first round but Dyas proved too strong, winning through in just over an hour to book a potential meeting with top seed Cibulkova in the quarter-finals.
There were to be no more upsets for Ukrainian qualifier Lyudmyla Kichenok either as she was beaten 6-1 6-2 by Poland's Magda Linette.
The 21-year-old beat eighth seed Ayumi Morita in the first round but was no match for Linette whose nine-game winning run across the sets ensured her a quarter-final berth.
She will face second seed Zhang Shuai in the last eight provided she can overcome wildcard Zhang Ling on Thursday.