"Trump called Erdogan tonight (Monday) and congratulated him on his success in the referendum", Turkish presidential sources said, quoted by the government run Anadolu news agency.
"The crusader mentality in the West and its servants at home have attacked us", he said.
According to unofficial results, the "yes" campaign won with 51.41 percent, while the "no" votes stood at 48.59 percent.
The queues formed as the main opposition party was scheduled to formally request that the electoral authority annul the referendum over the ballots lacking the official stamp. The HDP, for its part, claims votes from about 1,000 ballot boxes, representing as many as 250,000 voters, weren't accurately entered into the official tally. This work has now finished. "It is wrong to speak after the people have spoken".
Prime Minister Binali Yildirim, whose job will cease to exist once the constitutional changes take full effect, said Erdogan would be invited to rejoin the ruling AK Party as soon as official results are announced, a sign the government has no intention of waiting to see the outcome of opposition appeals.
Earlier, some Turkish media reported that the Russian leader had sent a telegram of congratulations to his Turkish counterpart.
On Monday, the head of Turkey's electoral board, Sadi Guven, strongly defended his decision to allow the controversial ballots, citing high demand for ballots and saying similar procedures had been followed in the past.
Much as democratic voters in the United States awoke in shock to realize the rural "flyover" states had carried Trump to victory, all of Turkey's major cities - Istanbul, Ankara, Izmir - voted hayir ("no") in the referendum.
"I suspect the result was narrower than what Erdogan expected", said Howard Eissenstat, associate professor of Middle East History at St. Lawrence University in Canton, New York.
The European Union opposed Erdogan's bid to shift the country to a system giving the president sweeping new powers. It allows the victor of the 2019 presidential election to take full control of government, ending the current parliamentary system and abolishing the role of prime minister.
The increasing polarization of Turkish society has long anxious observers, who note the dangers of deepening societal divisions in a country with a history of political instability.
The European Union on Tuesday urged Turkey to launch "transparent investigations" into alleged voting irregularities in the referendum granting President Recep Tayyip Erdogan extra powers.
Toner added that the US government was looking "to the government of Turkey to protect the fundamental rights and freedoms of all its citizens - regardless of their vote on April 16 - as guaranteed by the Turkish constitution and in accordance with Turkey's worldwide commitments, such as under the Helsinki Charter".
While German Chancellor Angela Merkel has called on the Turkish authorities to "seek respectful dialogue with all political social forces in the country", warning both sides against escalation in a timid statement, US President Donald Trump was one of the first western leaders to make a congratulatory call to Erdogan on Monday.
The White House readout of Mr Trump's call noted the pressing issues on which the USA has tried to work with Turkey, namely fighting the so-called Islamic State group and quelling Syria's civil war.