In Saturday's first round, Mr Fico polled 28.2% with Mr Kiska on 24% on a turnout of 43.4%.
Independent conservative Radoslav Prochazka was third with 20.8%.
As no candidate gained 50% of the votes, a second round run-off will be held in two weeks' time for the mainly ceremonial post.
Mr Fico's left-wing Smer party won the parliamentary election in 2012.
The 49-year-old had previously served as prime minister from 2006-10, pursuing an anti-austerity agenda.
Since the 2012 elections his party has governed alone - the first time since independence that a party secured an absolute majority in the Slovak parliament.
Slovakia's president has the power to appoint the prime minister, as well as the main figures in the judiciary.
However, parliament exercises legislative power.
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