In a phone call, the US president also thanked Mr Erdogan for supporting America's missile strike on a Syrian government airbase on 7 April.
In the poll, 51.4% of Turkish voters backed the changes.
Mr Erdogan rejected criticism from international monitors who said he had been favoured by an "unequal campaign".
"Know your place," he told the observers.
The narrow victory was ruled valid by Turkey's electoral body, despite claims of irregularities by the opposition.
Addressing supporters in the presidential palace in Ankara, Mr Erdogan said that Turkey did not "see, hear or acknowledge the politically motivated reports" of the monitors.
The result, he said, ended the debate on changing the constitution and creating an executive presidency, adding that the process of implementing the reforms would now begin.
He also said the country could hold a referendum on its long-stalled EU membership bid.
Additionally, Mr Erdogan said he would approve the death penalty if it was supported in a referendum or a bill was submitted to him through parliament. This would end Turkey's EU negotiations.
Earlier, Turkey's foreign ministry said the remarks by the monitors lacked impartiality. "Saying the referendum fell below international standards is unacceptable," it said in a statement.
The main opposition Republican People's Party (CHP) has demanded a recount of 60% of the votes. Its deputy head said the result should be annulled altogether.
The pro-Kurdish Peoples' Democratic Party (HDP) also challenged the vote.
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