Mr Zeman, 73, is seeking a second five-year term but has stoked controversy with his outspoken views and pro-Russian stance.
The second round is scheduled for 26-27 January.
The BBC's Rob Cameron in Prague says Mr Zeman appears to have done worse than expected in rural areas, where his forthright views on immigrants and political correctness have traditionally struck a chord. To avoid a run-off, he needed to secure more than half the vote.
Mr Drahos knows he has a good chance of unseating Mr Zeman when the two men go head to head in two weeks' time, he adds.
Many voters who chose one of the other seven candidates are likely to switch their allegiance to Mr Drahos in the second round.
President Zeman has become one of the EU's most outspoken opponents of sanctions against Moscow and has also made improving relations with China a priority.
Mr Drahos, a pro-European academic, has been forthright in his opposition to the president, saying: "We say in Czech that 'the fish stinks from the head' and that perfectly sums up Mr Zeman's term."
The first round of voting, which took place over two days, passed uneventfully apart from a semi-naked protester who tried to disrupt Mr Zeman casting his vote in Prague on Friday and accused him of being in the pocket of Russian President Vladimir Putin.
She was bundled away and Mr Zeman then cast his ballot, but was visibly shaken by the incident.
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