Mr Erdogan says the result is a "milestone for democracy"
The US and European Union have welcomed the result of the Turkish constitutional referendum.
Voters in Turkey gave strong backing to a package of changes to the country's military-era constitution.
The changes are aimed at bringing Turkey more in line with the EU, which the government wants to join.
This result will help PM Recep Tayyip Erdogan, who made the reform a test of his leadership, ahead of elections next year, correspondents say.
With nearly all votes in the referendum counted, about 58% had voted "Yes" to amending the constitution.
Mr Erdogan said the result meant the country had "crossed a historic threshold toward advanced democracy and the supremacy of law".
The European Commission has welcomed the results.
"As we consistently said in the past months, these reforms are a step in the right direction as they address a number of long-standing priorities in Turkey's efforts towards fully complying with the accession criteria," Enlargement Commissioner Stefan Fuele said in a statement.
Mr Obama called Mr Erdogan to congratulate him.
The US president "acknowledged the vibrancy of Turkey's democracy as reflected in the turnout for the referendum that took place across Turkey today", a statement released by the White House said.
German Foreign Minister Guido Westerwelle said the vote was critical for Turkey's bid to join the European Union.
"This discussion in society, also about the concrete form of the balance of power in the state, is very much to be welcomed," Mr Westerwelle said in a statement.
The result gives a boost to Mr Erdogan's political standing, after a hard-fought campaign in which the main opposition accused him and his party, the AKP, which has its roots in political Islam, of trying to take control of the judiciary, says the BBC's Jonathan Head in Istanbul.