Ince, head of the Republican People's Party and the main opposition presidential candidate, urged monitors and citizens to protect ballot boxes on voting day against fraud.
With more than half of ballot boxes counted Sunday, in the lead, Erdogan had 56.5 percent of the vote, which would be enough for him to win outright and avoid a runoff election.
In Sunday's parliamentary contest, the Islamist-rooted AK Party won 42 percent and its MHP ally 11 percent, based on 99 percent of votes counted, broadcasters said.
Results being compiled by the Fair Election Platform, formed by opposition parties, also pointed to Erdogan winning the presidency in the first round with about 53 percent. The president, however, is facing a more robust and united opposition, which has vowed to return Turkey to a parliamentary democracy with strong checks and balances.
In a tweet earlier in the evening, he said only 37 percent of ballot boxes had actually been counted, as opposed to the more than almost 90 percent Anadolu was reporting at the time. He had 7 per cent, based on 90 per cent of votes cast.
From a modest background himself, Erdogan has presided over an infrastructure boom that has modernized Turkey and lifted many out of poverty while also raising Islam's profile, for instance by lifting a ban on Islamic headscarves in schools and public offices.
And in a situation labelled as blatant unfairness by activists, the HDP's Demirtas has campaigned from a prison cell after his November 2016 arrest on charges of links to outlawed Kurdish militants.
Under constitutional amendments approved after a controversial 2017 referendum, Turkey is making a transition from a parliamentary system to a presidential one - giving the next president expanded powers, abolishing the prime minister's post, and eliminating numerous checks and balances created to help parliament protect against the misuse of presidential powers.The changes will take effect after the elections.
While Mr Erdogan met crowds of cheering supporters, the opposition has cried foul, claiming that state media and the election commission manipulated the results.
Voters are flocking to polling centres today to cast ballots in an election that will complete Turkey's transition to a new executive presidential system, a move approved in a controversial referendum past year.
The CHP said it had recorded violations in particular in the southeastern province of Sanliurfa, although Erdogan insisted there was no major problem.
There were reports of a scuffle at the polling station in Suruc and voting was briefly halted there.
Turkish election: Where has Erdogan's "economic miracle" gone? Emergency rule allows the government to bypass parliament.
Erdogan blamed the coup on his former ally, USA -based Muslim cleric Fethullah Gulen, and has waged a sweeping crackdown on his followers in Turkey, detaining some 160,000 people, according to the United Nations.
Erdogan says his tough measures are needed to safeguard national security.