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).