Voters were being asked to endorse an 18-article reform package put forward by the ruling Justice and Development Party that would replace the current system of parliamentary democracy with a powerful executive presidency.
Amid tight security, Erdogan cast his vote at a school near his home in Istanbul, surrounded by members of his family. He hoped Turks would make the "expected" choice, he said.
Polls closed at 5pm local time and results were expected later.
Earlier in the day, three people were reported to have died after an exchange of gunfire near a polling station in the southeastern province of Diyarbakır. The cause of the clash was not immediately clear and it was not known if there was a connection with the referendum.
Kemal Kilicdaroglu, leader of the one of the main opposition parties, the Republican People's Party (CHP), voted in Ankara along with his wife and son. "I hope the result will be good and then we will talk about the main problems of Turkey," he told CNN Turk.
Turkey's Prime Minister Binali Yildirim, who voted in Izmir, told reporters: "Whatever the result may be, it will be highly respected by us."
If passed, the new measures would amount to the biggest constitutional upheaval for the country since its foundation in 1923 after the demise of the Ottoman Empire.
They would cement Erdogan's grip on a country whose divisions have deepened since a failed coup attempt last July that ended with the deaths of more than 250 people and led to a the imposition of a fierce crackdown on dissent.
Erdogan said he believed Turks would vote with "common sense", adding: "Hopefully once the polls are closed and votes are counted, Turkey will march towards the future by making the expected choice."
Evoking the revered founder of modern Turkey, he said: "This referendum is no ordinary vote ... We are realizing the dream of Mustafa Kemal Ataturk to make Turkey a more civilized country."
Erdogan's supporters say the reforms will help kick start a lethargic economy and stabilize a nation dealing with the resurgence of a 30-year conflict with militants from the Kurdistan Workers' Party (PKK). Opponents fear the proposals amount to a constitutional dictatorship.
"I think this referendum is very important because this country had a lot of problems with the current system," 27-year-old Ozcan Ayata told CNN in Ankara Sunday after voting at a local school in the Sogutozu district.
"The tension between the President and the Prime Ministers during coalition governments caused crises that took the country 20 years back and allowed Europe to meddle in Turkey's interior politics."
But opponents worry about giving more control to a president already pushing the boundaries of his power. "My vote means my independence and emancipation," Gulsum Kilic, a 48-year-old interpreter from Ankara, told CNN.
"I voted for a good future for my children and everyone's children. Staying as a secular republic is very important for this country."
Another voter, Gurcan Hotoglu, said the outcome of the referendum could take the country back 200 years and "establish a dictatorship."
Erdogan's rise to power
After serving as prime minister for more a decade, Erodgan became president in 2014. By little more than force of personality, he turned a largely ceremonial post into a vehicle of significant power.
He has initiated a widespread crackdown on opposition, which intensified after last year's botched coup, for which he blamed US-based cleric Fethullah Gulen.
Critics say the clampdown has gone beyond the supporters of the coup and is designed to silence dissent in the runup to the referendum.
The arrests of 47,155 government critics, academics, journalists, military officials and civil servants have draw widespread international condemnation and strained Turkey's relations with the European Union.
If Turkey approves the new constitution, the new powers it accords to the president would kick in over time. The post of Prime Minister would be abolished after the 2019 national elections, and time-limits on the newly-empowered presidency would be reset. If he won in 2019 and 2024, Erdogan could be in power until 2029.
Erdogan would once again be able to lead the Justice and Development Party -- at the moment, the presidency is supposed to be politically neutral. He would also be able to appoint a number of senior judges.