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.