The 2008 champion had to take a medical time-out between the second and third sets for treatment on an undisclosed injury. She was grimacing when bending over and grabbing her lower back during the final set.
Sharapova's demise comes a day after Serbia's Ana Ivanovic beat world number one and title favourite Serena Williams, also in the fourth round.
"I watched that match, it was a great match for Ana, but this was a little bit of a different story," Cibulkova said in a courtside interview.
"The most important thing was that I went on the court and I was 100 percent sure I could win this match. I wasn't doubting myself."
Sharapova took the first set 6-3 as both players struggled with their serve, but Cibulkova won the second after racing to a 5-0 lead, which the Russian pegged back, before the diminutive Slovak finally held serve to send the match to a decider.
Sharapova then took the medical time-out but was unable to halt Cibulkova's momentum as the Slovak raced through to win the set and reach her first Australian Open quarter-final.
She will meet either eighth-seed Jelena Jankovic or 11th-seeded Romanian Simona Halep for a place in the last four.
"I just kept going. I knew what I had to do and I was doing exactly the right things," Cibulkova added.
"The most important thing is to believe in yourself."
(Reporting by Greg Stutchbury in Wellington; Editing by Peter Rutherford)