Thirty candidates are vying to replace interim leader Catherine Samba-Panza.
UN peacekeepers are patrolling to stop a repeat of the clashes during a recent referendum on a new constitution.
CAR has been torn by sectarian violence since a largely Muslim Seleka rebel group seized power in March 2013.
A band of mostly Christian militias, called the anti-Balaka, then took up arms against the Seleka.
In January 2014, a transitional government was formed but elections have been postponed four times since February 2015 due to insecurity and logistical challenges.
‘We want to be free’
Three candidates are considered frontrunners for the presidency.
Two of them – Martin Ziguele and Anicet Dologuele – were prime ministers under late President Ange-Felix Patasse. The third, Karim Meckassoua, is a Muslim who served as a minister under President Francois Bozize until he was ousted in 2013 by Seleka rebels.
UN peacekeepers staged patrols and positioned armoured personnel carriers at polling stations in the capital, Bangui, Reuters news agency quotes witnesses as saying.
“We came to vote because we want to be free to get back to our professions. We want to put an end to the conflict,” said businessman Gradias Vara.
However, many polling stations, including in the flashpoint PK5 district, failed to open on time because ballot papers and voters’ lists had not yet arrived, the AFP news agency reports.
Pope Francis ignored safety fears to visit CAR last month, and said Muslims and Christians were brothers and sisters who should live in peace.
Observers say a second round is likely and expect it to be held by the end of January.
Elections are also taking place for the 105-seat National Assembly.
After seizing power, the Seleka rebels installed Michel Djotodia as the first Muslim leader of the majority Christian country.
But under pressure from regional leaders and former colonial power France, Mr Djotodia stood down and was succeeded by Ms Samba-Panza.
Mr Djotodia and Mr Bozize are both in exile and face UN and US sanctions over violence in the country.
CAR’s Constitutional Court rejected Mr Bozize’s bid to contest the election.
He described the decision as shameful.
More than 1.8 million people are registered to vote, out of a population of roughly five million.
Some one million people have been forced to flee their homes because of the fighting.
Source: BBC
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