If no presidential candidate wins more than 50% in the June 24 vote, a second round run-off will be held on July 8.
However the opposition raised doubts about the accuracy and reliability of the figures released by state-run Anadolu news agency, the sole distributor of the official vote tally.
The stakes in this election are particularly high as the new president will be the first to enjoy enhanced powers under a new constitution agreed in a April 2017 referendum strongly backed by Erdogan.
It has condemned what it calls Mr Erdogan's "one-man rule".
The report added that Turkey's main opposition Republican People's Party (CHP) has won 14.82 percent of the vote while voter turnout has been put at 86.82 percent for presidential and 87 percent for parliamentary elections.
Polls show Erdogan falling short of a first-round victory in the presidential race.
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In the parliamentary election, a "People Alliance" consisting of Erdogan's ruling party and a nationalist party led at around 59% with about 36% of ballot boxes counted.
She and others in the city said they voted for the pro-Kurdish HDP, hoping it would exceed the 10 per cent threshold of votes needed to enter parliament.
There is also a Kurdish presidential candidate, Selahattin Demirtas, who is running from behind bars after being imprisoned in November 2016 as part of the purge following the attempted coup.
He is now leading in the polls with 56 percent of the vote.
AK Party supporters clogged a main road in the capital Ankara leading to party headquarters honking horns in celebration.
Polling stations opened at 0500 GMT (10.30 IST) and were due to close at 1400 GMT (7.30pm IST), with the first results expected late in the evening.
Opposition parties and NGOs deployed up to half a million monitors at ballot boxes to ward against possible electoral fraud.
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Over 56 million eligible voters were for the first time casting ballots in both elections, with Erdogan looking for a first round knockout and an overall majority for his ruling Justice and Development Party (AKP) to extend his 15 year grip on power.
"It's time for change", said Aynur, 40, an architect who cast her vote at a polling station near Istanbul's central Taksim Square.
Erdogan counters that view, saying "Turkey is staging a democratic revolution". "With the presidential system, Turkey is seriously raising the bar, rising above the level of contemporary civilizations".
Erdogan, the most popular but also divisive leader in modern Turkish history, argues the new powers will better enable him to tackle the nation's economic problems - the lira has lost 20 percent against the dollar this year - and crush Kurdish rebels in southeast Turkey and in neighbouring Iraq and Syria.
He wants to scrap the post of prime minister and weaken parliament, bringing in a new executive presidency instead.
Although Erdogan dominated airtime on a pliant mainstream media, Ince finished his campaign with eye-catching mass rallies, including a mega meeting in Istanbul on Saturday attended by hundreds of thousands.
Almost 300 Turkish diplomats have sought asylum in Germany since a failed coup in Turkey in July 2016, a German newspaper reported on Saturday. This state restricts some freedoms and allows the government to bypass parliament with decrees. Erdogan accused his late ally and now nemesis, US-based cleric Fethullah Gulen, of masterminding the coup.
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The president's critics, including the European Union which Turkey still nominally aspires to join, say Erdogan has used the crackdown to stifle dissent.