The referendum allows the Islamist-leaning president to have the power to dissolve Turkey's parliament and have control over his country's judiciary.
"President Trump thanked President Erdogan for supporting this action by the United States, and the leaders agreed on the importance of holding Syrian President Bashar al Assad accountable", the statement continued.
He hit back at threats by European Union leaders to freeze accession talks, adding: "We will sit down and talk, and we can hold a referendum for that (EU bid) too!"
Election authorities said preliminary results showed 51.4 percent of voters had backed the biggest overhaul of Turkish politics since the founding of the modern republic.
But the opposition immediately cried foul, claiming a clean vote would have made a difference of several percentage points and handed them victory.
Opposition parties indicated that they are encouraged by the results, which gave them effectively half of the votes, and that they consider the referendum tainted by irregularities.
Bulent Tezcan, deputy head of the CHP demanded the referendum be reheld, saying that would be the "only decision that will end the debate about the legitimacy" and ease people's concerns.
"This referendum will forever remain controversial", he told reporters.
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But Mr Erdogan has taken his authoritarian bent to an extreme, imposing the state of emergency and purging the opposition, academia and the army after a failed coup past year.
But opponents fear the changes will lead to autocratic one-man rule, ensuring that Mr Erdogan, who has been accused of repressing rights and freedoms, could govern until 2029 with few checks and balances. Opposition parties are contesting the results, objecting to a decision by the election board to lift a rule requiring ballots to have official seals and citing discrepancies between vote totals released by the election board and a state-owned news agency. They set a limit of two five-year terms for presidents.
Erdogan earlier congratulated cheering supporters at Ankara's airport for "standing tall" in the face of the "crusader mentality" of the West.
About 2,000 protesters gathered in Istanbul's Besiktas neighborhood on Wednesday evening over the referendum approving constitutional changes sought by President Recep Tayyip Erdogan.
If Turkey were to hold a referendum on bringing back capital punishment it would be a break with European values, the Hollande's office warned. "He imposed a very restrictive environment for the "no" camp", said Fadi Hakura, a Turkey specialist at the London-based Chatham House think tank.
Turkey's new political system is due to come into effect after elections in November 2019, although Erdogan is expected to rapidly rejoin the ruling Justice Development Party (AKP) he founded but had to leave when he became president.
Erdogan insists the reforms will create a fast and efficient system of governance that will allow Turkey to face the challenges of fighting terror and the slowing economy.