Samantha Stosur won her second-round clash against Barbora Strycova but was unable to back it up later in the day while top seed Angelique Kerber was bundled out as rain continued to play havoc with the Qatar Open schedule.
Stosur led eighth seed Strycova 1-0 when their match was delayed by rain on Wednesday. She took the first set 6-3 before losing the second by the same margin and recovering to take the third 6-4.
The former US Open champion then pushed third seed Cibulkova in both sets of their clash but was unable to break through, going down 7-5 6-4 in what was the Slovakian's 400th career win and first over Stosur.
"She was really fighting and really into the match," Cibulkova told metro.co.uk.
"It came down to a few points here and there; you can see from the scoreline that it was very tight.
"I tried to have a new tactic against Sam because I'd never beaten her before, and it was really working today."
Kerber suffered a shock 6-4 0-6 6-4 defeat by Russian teenager Daria Kasatkina.
The 19-year-old, ranked 32 in the world, survived six break-point chances as she took the opening set.
Kerber, the world No.2, rallied to level the match after breezing through the second set in less than 30 minutes.
However, Kasatkina responded with two early breaks in the decider to lead 3-0 and then closed out victory.
Her quarter-final clash with Monica Puig, tied at one set all, was delayed by rain on Thursday evening as was the match between former world No.1 Caroline Wozniacki and Lauren Davis.
