Zeman, 73m was elected in 2013 during the country's first direct presidential vote, a victory that returned the former left-leaning prime minister to power.
Although Zeman was in pole position Saturday, the outspoken head of state was unlikely to win an outright majority, with a run-off expected on January 26-27.
"I voted for professor Drahos because I want that someone who will not push us to the East and who will not be a disgrace", said lawyer Matej Gredl, 30, after he voted in Prague.
Zeman and Drahos, the former head of the Czech Academy of Sciences, advanced to the second round scheduled for January 26-27 because none of the nine presidential candidates received a majority of votes in the first round.
As a member of an increasingly right-wing regional alliance of Central European nations, named the Visegrad Group, that includes Hungary, Poland and Slovakia, the Czech Republic stands at an ideological crossroads with the rise of the country's xenophobic far right, which rose to parliament in October's legislative elections. First round results are due out later Saturday.
A recent poll for Czech Television showed Drahos winning a second round vote with 48.5% support against 44% for Zeman. Among them, Jiri Drahos is the most likely candidate to advance.
In a 2015 Christmas message, Zeman described the ongoing influx of migrants in Europe as an "organized invasion".
Security personnel also had to help a visibly rattled Zeman, who walks with a cane, to leave the room. A mild-mannered liberal centrist whom critics have dubbed "wishy-washy", he has called for Prague to "play a more active role in the EU" and has backed the adoption of the euro. He has criticised migration from Muslim-majority countries as well as Germany's decision to accept many migrants.More news: Mark Helfrich reportedly hired as Chicago Bears new OC
"The president should work to unite society".
Zeman's rhetoric and actions has provoked the ire of many Czechs, particularly in the capital, Prague, where Zeman cast his ballot Friday.
"Polarisation of society has deepened in the past months", Saradin said.
Zeman says he is ready "to meet (Drahos') request" to face each other.
The outcome may influence Czech Prime Minister Andrej Babis's chances of finally forming a cabinet as his first attempt to rule in a minority administration is likely to be rejected by parliament next week. In addition to Zeman - a heavy smoker and proud drinker - is Czech singer Petr Hannig and former prime minister Mirek Topolanek, who in 2009 was famously photographed naked with topless women at one of former Italian Prime Minister Silvio Berlusconi's villas.
Mr Zeman's forthright views on immigrants and political correctness have struck a chord in rural communities, our correspondent says.
A former center-left prime minister, Zeman has warm relations with Russian President Vladimir Putin and has called for the removal of European Union sanctions imposed over Russia's annexation of Ukraine's Crimea.
But the situation could change dramatically if Drahos wins. "I'm still young, full of strength and full of energy and I look forward to debate", he remarked, while urging supporters to bring along their close ones, including "relatives and lovers", to the second-round polling stations.