With 99.45% of ballots counted, the "Yes" campaign had won 51.37% and "No" 48.63%, and the electoral board called victory for "Yes".
Erdogan supporters say replacing the parliamentary system with an executive presidency will modernise the country.
Turkey's two main opposition parties said they would challenge the results.
The Republican People's Party (CHP) demanded a recount of 60% of votes. They criticised a decision to accept unstamped ballot papers as valid unless proven otherwise.
As jubilant Erdogan supporters rallied in the big cities, pots and pans were banged in Istanbul by opponents of the referendum, in a traditional form of protest.
Three people were shot dead near a polling station in the south-eastern province of Diyarbakir, reportedly during a dispute over how they were voting.
The European Commission called on the Turkish authorities in a statement to "seek the broadest possible national consensus" when implementing the constitutional reforms.