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.
In a separate development, Turkey extended the state of emergency for three months. The measure, introduced after a failed coup last July, was set to expire in two days.
In a statement on Monday, the White House said President Trump discussed with his Turkish counterpart America's "action in response to the Syrian regime's use of chemical weapons" on 4 April.
It said the two leaders "agreed on the importance of holding Syrian President Bashar al-Assad accountable".
They also discussed "the counter-ISIS [Islamic State campaign]", the statement added.
Relations between Washington and Ankara have recently been strained by several key issues.
One of Turkey's main grievances with the US is the policy started by the Obama administration of supporting Kurdish fighters in Syria who are fighting IS forces.
Turkey views the Syrian Kurdish People's Protection Units (YPG) as a terror group linked to Kurdish separatists waging an insurgency inside Turkey since 1984.
The two sides are also at loggerheads over Fethullah Gulen. Turkey accuses the Pennsylvania-based cleric of orchestrating the failed coup and wants him extradited.
Officially Washington insists any decision on returning him to Turkey from the US remains a judicial rather than a political one.
Despite saying that the voting day was "well administered", the Organization for Security and Co-operation in Europe (OSCE) and the Council of Europe criticised the referendum campaign, saying:
- It was an "unlevel playing field" and the two sides of the campaign "did not have equal opportunities"
- It was unbalanced due to the active involvement of the president and several senior officials
- It was tarnished by a number of officials equating No supporters with terrorist sympathisers
- State resources were misused
- Under the state of emergency, essential fundamental freedoms were curtailed
- Despite some measures, the legal framework remained inadequate for a genuinely democratic referendum
"In general, the referendum did not live up to Council of Europe standards," said Cezar Florin Preda, head of the Council of Europe delegation.
The council is a pan-European human rights body of which Turkey is a member.
The monitors also criticised a late change by electoral officials that allowed voting papers without official stamps to be counted. They said this move "removed an important safeguard and were contested by the opposition.
But the head of Turkey's electoral body, Sadi Guven, said the unstamped ballot papers had been produced by the High Electoral Board and were valid. He said a similar procedure had been used in past elections.
Turkey reacts: 'Politically motivated reports'
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.
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.
European leaders concerned
In Europe, leaders have been less than enthusiastic in their comments over the result.
German Chancellor Angela Merkel warned that the "tight referendum result shows how deeply divided Turkish society is and that means a big responsibility for the Turkish leadership and for President Erdogan personally".
The European Commission issued a similar call.
Others expressed concern about the possibility of the return of capital punishment.
The French president's office warned that any referendum on reviving the death penalty would "obviously be a break with values and engagements" that Turkey had accepted in joining the Council of Europe. The president of the European Parliament, Antonio Tajani, tweeted his own concerns.
Austrian Foreign Minister Sebastian Kurz went further. He said the referendum result was a "clear signal against the European Union". The "fiction" of Turkey's bid to join the bloc must be ended, Mr Kurz said.
What do the constitutional changes include?
- The president will have a five-year tenure, for a maximum of two terms
- The president will be able to directly appoint top public officials, including ministers and one or several vice-presidents
- The job of prime minister will be scrapped
- The president will have power to intervene in the judiciary, which Mr Erdogan has accused of being influenced by Fethullah Gulen, the Pennsylvania-based preacher he blames for the failed coup in July
- The president will decide whether or not impose a state of emergency.