The party leaders moved in after an opposition referendum on holding an early election failed to attract enough voters on Saturday.
Only 27.3% of voters cast ballots, far below the absolute majority a referendum must attract to be valid, according to data from the Statistics Office.
An early election can be held in Slovakia if the constitution is changed to permit parliament's four-year term to be shortened. Such a change of constitution would need approval by a referendum or by at least 90 votes in the 150-seat parliament.
The government of Prime Minister Eduard Heger has been ruling in a caretaking capacity after it lost a no-confidence vote in parliament in December.
Heger's government, criticised for not doing enough to help people hit by soaring energy bills and inflation, fell in a no-confidence vote in December.
Its former ruling partner, the libertarian Freedom and Solidarity (SaS) party, had joined opposition parties in the motion to defeat the government, which was in a minority since SaS quit the coalition in September.
Heger had hoped to form a new majority and rule until the next election scheduled in February 2024, but SaS refused to negotiate, leaving him with few options to stay on.
(Reporting by Michael Kahn, Editing by Hugh Lawson)
