Twentieth-seed Cibulkova won 4-6, 6-3, 6-1 to reach the quarter-finals for the first time. She will take on Romania's Simona Halep, the 11th seed who beat Croatia's Jalena Jankovic.
An injury-hit Sharapova won the first set but failed in the next two
Sharapova, the champion of 2008, was plagued by a hip problem an made 45 unforced errors in the match, served eight double faults and was broken seven times.
Cibulkova, 24, said after the match that she had no doubt about a win. "I was 100 per cent sure I could win," she said.
Sharapova had won three of the previous five meetings against Cibulkova. She also looked favourite to win Monday's encounter by winning the first set. But the Slovakian came back strongly in the second and third sets.
Defending champion Victoria Azarenka beat USA's Sloane Stephens 6-3, 6-2.