After splitting the singles matches, Cibulkova and Hrbaty rallied to beat Casey Dellacqua and Lleyton Hewitt 6-7 (5), 6-3, 10-4 in the deciding mixed doubles match.
It was the second upset in as many nights at the international mixed team tournament for the Slovaks, who beat the top-seeded United States 3-0 on Monday night.
Australia lost its opening match to Germany and cannot reach the final from Group A.
In Group B, Italy stayed in contention with a comeback 2-1 win over France. Russia, which opened with a 2-1 win over the Italians, is against Taiwan on Wednesday.
Slovakia's two upsets have started with wins by Cibulkova.
She beat Dellacqua 7-5, 6-2 on Tuesday night to give her team a 1-0 buffer.
The Australians worked back into calculations when Hewitt finally overcame Hrbaty 7-6 (4), 6-7 (1), 6-4.
Hewitt was two points from victory in the ninth game of the second set before being forced into a third by Hrbaty's comeback in the almost 3-hour match.
Former top-ranked Hewitt is only two singles matches into a comeback after a five-month layoff with a hip problem.
But his two singles wins have been overshadowed by Hrbaty, whose ranking has slipped to 250 since he had an operation on his right shoulder to remove a bone spur late in 2007.
Earlier Tuesday, Simone Bolelli beat Gilles Simon in straight sets to cancel out France's advantage, then combined with Flavia Pennetta to guide the Italians to a win.
When Alize Cornet gave France a 1-0 lead with a 7-5, 6-2 win over Pennetta in the women's singles, Italy was only one loss from slumping out of contention.
But Bolelli beat No. 7-ranked Simon 6-3, 6-3 in the men's singles and then helped Pennetta in a 6-4, 7-6 (2) mixed doubles win.
"Finally I won against a top 10. First time,'' Bolelli said. "I played very well, I served very good. I played aggressive .... I played almost perfectly I think.''
France opened with a 3-0 win over Taiwan. Now there's three teams in the group with a one win, although Russia's brother-sister team of Marat Safin and Dinara Safina has a match in hand.
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