But predictably, state media announced Mr Erdogan as the victor earlier on - saying he had won 53 percent of 99 votes cast, beating nearest rival Mr Ince on 31 percent.
Several world leaders supportive of Erdogan, including Azerbaijani President Ilham Aliyev and Qatari Emir Tamim bin Hamad Al-Thani, called to congratulate him on his "victory", the presidency said.
Erdogan spoke of his commitment to fight terrorist organisations and "to continue the fight to make the Syrian grounds freer" and to better the country's "international reputation".
Erdogan's victory could also add to Turkey's division with the North Atlantic Treaty Organization.
Ince, of the secular Republican People's Party (CHP), conceded defeat in a news conference in Ankara, but railed against the vote, which he said was unfair and would have serious consequences for Turkey's democracy.
"We're now in a one-man rule - there's no mechanism to prevent arbitrary rule".
AKP's ally the MHP (Nationalist Action Party) received 11.1 percent of the vote and obtained 49 seats in parliament, compensating somewhat for AKP's poor performance.
Amid signs of a weakening economy, Erdogan in April declared that presidential and parliamentary elections would be held on June 24, 17 months earlier than planned.
For one thing, the Turkish economy is doing well now, a boon to the Erdogan administration that might not last another year.
The $20 billion Akkuyu nuclear power plant will be the first in Turkey.
In a series of Twitter postings, Demirtas praised the party's success in winning a projected 67 seats out of 600, according to unofficial results.
"The nation has entrusted to me the responsibility of the presidency and the executive duty", the 64-year-old Erdogan said in televised remarks from Istanbul.
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With election victory in hand, Erdogan will rule with new muscle provided by sweeping changes to the constitution that were approved by a narrow majority of the electorate in a controversial referendum that he championed past year.
Trailing were Selahattin Demirtas of the pro-Kurdish Peoples' Democratic Party (HDP) with over eight percent in third and Meral Aksener of the nationalist (Iyi) Good Party with over seven percent.
The paper also alleges that on election day, "there were reported cases of fraud, including ballot stuffing, [such as] an incident where a auto filled with ballots was pulled over heading to a polling station near the southern city of Urfa".
Hamas's political chief and the Palestinian Authority's president were among the first to congratulate the Turkish president on his election win.
"Erdogan and his team will earn experience and will try to correct the system's flaws while implementing it", he said, adding that there will be new decrees and legislations to achieve that.
Some 50,000 people have been arrested and more than 110,000 civil servants have been fired in a massive government crackdown that has taken place under a state of emergency imposed after the coup that is still in place.
The president told crowds in Ankara in the early hours of Monday morning: "This election's victor is democracy, this election's victory is national will".
The CHP's share of parliamentary votes, under the leadership of Kemal Kilicdaroglu, dropped by nearly 3 percentage points to 22.6 percent as compared to the results of the November 2015 elections.
He said: "Mr Erdogan is now an all-powerful man, not just de facto but also formally".
There was no word immediate reaction from the United States. As Turkey's democracy backslides, its European Union accession seems ever further away; however, Turkey will not want to lose European Union investment and trade and the West will not sacrifice the geostrategic importance of the country so relations, although perhaps strained, seem set to continue.
Ince's succeeded in securing 30.6 percent of the presidential vote, which is a first for any CHP contestant in decades.
Margot Wallstrom, Sweden's foreign minister, said Turkey's democracy had shortcomings - she cited opposition leaders sitting in jail - but said Erdogan should be given the chance to fix that.