Trump joins a short list of leaders who have openly congratulated Erdogan, including Azerbaijani President Ilham Aliyev and Saudi King Salman.
Votes are shown to observers at a voting station in Istanbul on Sunday.
.Erdogan argues that concentration of power in the presidency is needed to prevent instability.
The French government said it would "follow with great care" the global monitors' final report in coming weeks, particularly in relation to a reported last-minute change of rules by the electoral boards to allow ballots that had not been officially stamped.
The "Yes" camp won 51.41 percent in Sunday's referendum and "No" 48.59, according to complete results released by election authorities.
Turkey's main opposition began a battle on Tuesday to annul a referendum handing President Tayyip Erdogan sweeping new powers, while the bar association and an worldwide monitor said an illegal move by electoral authorities may have swung the vote. It was the third such extension since a failed coup attempt last July.
Opponents accuse him of leading a drive towards one-man rule in Turkey, a North Atlantic Treaty Organisation member that borders Iran, Iraq and Syria and whose stability is of vital importance to the United States and the European Union.
Trump and Erdogan also agreed on the importance of holding Syrian President Bashar al-Assad accountable for the chemical attack and discussed the campaign against the Islamic State, the statement said.
Supplier sues again over Arkansas execution drug
The original plan was to double the executions over four nights, prior to the state's supply of lethal injection drugs expired. Their second claim is that he suffers from an intellectual disability that stems from fetal alcohol syndrome.
After the result, Erdogan said he would reintroduce the death penalty if it was backed in a referendum or a bill was submitted to him through parliament, a move the European Union says would put paid to Turkey's long-stalled negotiations to join the bloc.
"We have said this over and over in my speeches".
The Organisation for Security and Cooperation in Europe and it's Office for Democratic Institutions and Human Rights had worldwide observers in Turkey to monitor the referendum.
Erdogan and the government have staunchly rejected the findings by the European election observers, calling them biased.
The pro-Kurdish opposition Peoples' Democratic Party said it presented complaints about unstamped ballots affecting 3 million voters, more than twice the margin of Erdogan's victory. "Everyone should respect the outcome, especially the main opposition".
"There is only one decision to ease the situation in the context of the law - the Supreme Election Board (YSK) should annul the election", the Dogan news agency quoted CHP deputy leader Bulent Tezcan as saying. The amendments to the constitution are likely to allow him to hold office for two more terms until 2029.
"These complaints are to be taken very seriously and they are, in any case, of such an extent that they would turn around the outcome of the vote", Alev Korun told ORF radio.
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 will be able to select and remove senior civil servants without parliamentary approval.
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