The poll took place amid a deepening economic crisis in the country and concerns among Croats over high-level corruption.
Ruling party slide
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.
Jospipovic, the 52-year-old law professor and music composer, is expected to go up against Milan Bandic, the populist mayor of Zagreb, or Nadan Vidosevic, a wealthy businessman and former member of the ruling HDZ party, in any run-off.
The conservative ruling party’s chosen candidate, Andrija Hebrang, a former health minister, is not expected to gain much ground due largely to discontent over unemployment.
Polling ended at 7pm (18:00 GMT), with preliminary results expected by midnight (23:00 GMT).
Croatia’s domestic presidential powers are 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 in the running stand accused of questionable past dealings, including Bandic and Vidosevic.