Both candidates have pledged to back Croatia's bid for European Union membership, which it hopes to achieve in 2012.
Twelve candidates were in the running to replace Stipe Mesic, a popular reformist who started the process of getting Croatia into the European Union.
"The results show that justice has won"
Social Democrat candidate
About 4.4 million people - including about 400,000 Croats living abroad - were eligible to vote.
The poll took place amid a deepening economic crisis in the country and concerns among Croats over high-level corruption.
Mesic is ineligible to stand for re-election, having already served two terms of five years.
He led his country into the Nato military alliance, but his EU ambitions, so far, remain unfulfilled.
"The results show that justice has won," said Josipovic, who ran his campaign under the slogan "Justice for Croatia," insisting on the need to fight corruption.
He called for voters in the run-off to "cast ballots for justice, a better and more rightful Croatia, for the light and not for the darkness."
"I believe that Croatia will choose the light also on January 10."
The domestic powers of the Croatian presidency are limited and largely ceremonial, with the main decisions being made by the cabinet and parliament.
But the president leads the army and wields strong influence over foreign policy.
Davor Gjenero, a political analyst, said the president "hasn't got the type of political powers to do any serious political damage".
"But the result of the elections could be quite shameful for Croatia" if the elected candidate has been blemished by scandals, he said.
The fight against corruption is a key condition for joining the EU, and the government has recently begun cracking down.
Several of the candidates who stood in the first round stand accused of questionable past dealings, including Bandic.