Erdogan said 25 million people had supported the proposal, which will replace Turkey's parliamentary system with an all-powerful presidency.
That appeared short of the decisive victory for which he and the ruling AK Party had aggressively campaigned.
"It's close enough that the opposition says it's going to challenge alleged irregularities", he says. "That is why it is very significant".
The new presidential system takes effect at the next election, now slated for 2019.
What new powers will the president have?
Deputy Prime Minister Mehmet Simsek said there would be no early elections following the result.
"Under the state of emergency put in place after the July 2016 failed coup attempt, fundamental freedoms essential to a genuinely democratic process were curtailed".
The head of the main opposition Republican People's Party (CHP), Kemal Kilicdaroglu, said the legitimacy of the referendum was open to question. "We took this decision before one single vote was counted and registered into system". The involvement of Erdogan and other national and local public figures in the "yes" campaign led to a "restrictive" and "imbalanced" campaign framework, she said.
Opposition supporters took to the streets of Istanbul to bang pots and pans, a traditional form of protest, in a series of noisy demonstrations.
As the results came in, thousands of the president's supporters converged at the Ankara headquarters of the AKP, which Ergodan founded.Waving flags they shouted, "Tell us to kill, we will kill".
If Turkey were to hold a referendum on bringing back capital punishment it would be a break with European values, the Hollande's office warned.
Tezcan said any decision that changes Turkey's political system to such a vast extent should have been passed with an overwhelming endorsement. "It is of utmost importance to secure the independence of the judiciary in line with the principle of rule of law enshrined in the European Convention on Human Rights".
During the referendum campaign, Erdogan lashed out at Germany and the Netherlands, accusing them of acting like the Nazis when they barred rallies for the "Yes" camp. The new powers would also allow Erdogan to issue decrees, declare emergency rule, appoint ministers and state officials as well as dissolve the parliament while the prime minister's position would be abolished.
He reiterated his request to suspend European Union accession talks with Ankara.
Results were highly close to those of the latest polls before the referendum that have shown a slight difference in favor of the amendments' backers.
Throughout the campaign, Erdogan launched bitter attacks on the European Union, accusing member states of behaving like the Third Reich in failing to allow his ministers to campaign among expats.
The referendum has bitterly divided the nation.
Erdogan also thanked the "yes"- voters supporting political party leaders and the voters on the referendum victory.
Yet it will be hard for Erdogan to stabilise his country because he has previously specialised in provoking crises, such as the Kurdish insurgency since 2015 or Turkey's role in the war in Syria, which supposedly necessitate a strong leader such as himself. Yet even when I interviewed one of those deputies last week, he still insisted that it was the right thing to do to attend those funerals. 'Istanbul hayir diyor!' ('Istanbul says no!') reads one piece of stencilled graffiti, above the logo of the Turkish communist party.
"April 16 is the victory of all who said "yes" or 'no, ' of the whole 80 million, of the whole of Turkey", Erdogan told reporters in a live televised address.
While the French leader said "it's up to Turks, and them alone, to decide their political organization", he added that France will "follow with the greatest attention" outside observers' evaluations of whether the vote was free and fair.