Turkey's mainly Kurdish southeast and its three main cities, including the capital Ankara and the largest city Istanbul, looked set to vote "No" after a bitter and divisive campaign.
Erdogan said 25 million people had supported the proposal, which will replace Turkey's parliamentary system with an all-powerful presidency and abolish the office of prime minister, giving the "Yes" camp 51.5 percent of the vote.
That appeared short of the decisive victory for which he and the ruling AK Party had aggressively campaigned. Nevertheless, thousands of flag-waving supporters rallied in Ankara and Istanbul in celebration.
"For the first time in the history of the Republic, we are changing our ruling system through civil politics," Erdogan said, referring to the military coups which marred Turkish politics for decades. "That is why it is very significant."
Under the changes, most of which will only come into effect after the next elections due in 2019, the president will appoint the cabinet and an undefined number of vice-presidents, and be able to select and remove senior civil servants without parliamentary approval.
Erdogan himself survived a failed coup attempt last July, responding with a crackdown that has seen 47,000 people detained and 120,000 sacked or suspended from their jobs.
Kilicdaroglu has accused Erdogan of seeking a "one-man regime", and said the proposed changes would put the country in danger.
In some affluent neighbourhoods in Istanbul, people took to the streets in protest while others banged pots and pans at home - a sign of dissent that was widespread during anti-Erdogan protests in 2013.