But the scoreline did not reflect the fight Cibulkova put up against Great Britain's No.1.
An aggressive performance by the Slovakian put 26-year-old Konta under significant pressure, though she took the first set with two breaks of serve.
Despite breaking early in the second set, Konta was tormented by Cibulkova who fought hard, and had three break points taking a 4-2 lead, though Konta regained ground to break at 4-4, before serving out her victory.
She will meet Simona Halep in the quarter-finals, who beat number 15 Anastasija Sevastova 6-4 6-3.
Meanwhile, Caroline Wozniacki made it through to the quarter-finals after ending the run of qualifier Ashleigh Barty on Thursday night.
Wozniacki was a comfortable 6-4 6-2 winner as the sixth seed reached the last eight of this tournament for the fourth time in her career.
Barty was impressive early on, breaking Wozniacki to take a 3-1 lead, but it proved a temporary setback for the world number five.
Fourth seed Garbine Muguruza was given more of a workout as she saved match points to battle past American Madison Keys, beating the 16th seed 6-4 3-6 7-6 (7/3).
Eighth seed Svetlana Kuznetsova beat Spaniard Carla Suarez Navarro 6-2 6-4.
Karolina Pliskova remained on track to defend her title at 3-0 up in the first set against Italy's Camila Giorgi, as play was suspended for the day.
Additional reporting by PA Sport.
WTA Cincinnati 3rd Round
4-Garbine Muguruza Blanco (Spain) beat 8-Svetlana Kuznetsova (Russia) 6-2 5-7 7-5
Sloane Stephens (U.S.) beat Ekaterina Makarova (Russia) 2-6 6-3 6-4
Julia Goerges (Germany) beat 5-Elina Svitolina (Ukraine) 7-5 6-4
1-Karolina Pliskova (Czech Republic) beat Camila Giorgi (Italy) 6-3 4-6 6-0