Turkey's election board said the 'Yes' campaign secured victory with just over 51% of the vote.
The European Commission has called on President Erdogan to respect the closeness of the result and the far-reaching implications of the constitutional amendments.
Opposition parties have argued that the changes, which come into effect after the 2019 presidential election, give too much power to the office.
They have also demanded a recount claiming that there had been voting irregularities.
Relations hit a low during the referendum campaign when EU countries, including Germany and the Netherlands, barred Turkish ministers from holding rallies in support of the changes.
Mr Erdogan called the moves "Nazi acts" and said Turkey could reconsider ties with the European Union after many years of seeking EU membership.
Former Belgian prime minister Guy Verhofstadt, who heads the liberal group of MEPs in the European Parliament, said Mr Erdogan needed to change course, noting the result was very tight. "If Erdogan persists, EU should stop accession talks," he said.
After the vote, Mr Erdogan repeated his intention to review Turkey's suspension of the death penalty, a step which would almost certainly spell the end of Ankara's EU accession process.
***Voilà l'homme tout entier, s'en prenant à sa chaussure alors que c'est son pied le coupable. Ca devient inquiétant. Un des larrons fut sauvé. C'est un pourcentage honnête.