The Slovakian, who has never made it beyond the last eight, took the match 6-3 5-7 9-7 in a time of three hours on Court Three.
Radwanska lost serve in the fourth game of the opening set, which her opponent used as a platform to push on and race ahead in a match that went the distance.
Both players struggled to hold in the second set, taking it in turns to edge into the lead before Cibulkova squandered a match point to wrap things up.
Third seed Radwanska recovered well to level up the match at 1-1, although it would be her world number 18 opponent who somehow got over the line in a mammoth decider.
Yet another match point came and went in the 12th game, only for Cibulkova to keep her composure and break Radwanska for a seventh time in the match to get the job done.