The narrow victory was ruled valid by Turkey's electoral body, despite claims of irregularities by the opposition.
In contrast with the European Union attitude, Donald Trump congratulated the Turkish president personally on the phone on Monday.
Turkish President Recep Tayyip Erdogan has dismissed concerns a referendum on constitutional amendments did not live up to worldwide standards, telling voting monitors to "know your place". He will be allowed to make judicial and government appointments, issue decrees and declare state of emergencies.
The opposition has been particularly incensed by a last-minute move by the board to accept ballot documents in envelopes without an official stamp.
The opposition had immediately cried foul, and called for the vote to be annulled.
"We look 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 global commitments", the State Department said in a statement.
With the opposition crying foul over alleged violations, all eyes will be on Monday's announcement by global observers from the OSCE and the Council of Europe who will give their initial assessment of the vote.
In the final hours of voting on April 16, Turkey's High Election Board waived a legal requirement to invalidate ballots that aren't properly stamped, explicitly contravening Article 98 of Turkey's 2010 electoral law.
Bulent Tezcan, deputy head of the CHP demanded the referendum be reheld, saying that would be the "only decision that will end the debate about the legitimacy" and ease people's concerns.
Responding to the results, Mr. Erdogan said Turkey had made a historic decision in backing an executive presidential system, adding that the "Yes" camp had secured 25 million referendum votes, 1.3 million ahead of "No". Berlin said "take note" of the provisional result of the vote that saw the "yes" win with 51.37% after tabulation of the ballots in 99.45% of the ballot boxes. Everyone and all sections - and the main opposition party in particular- must show respect. Critics argue the reforms will hand extensive power to a man with an increasingly autocratic bent, leaving few checks and balances in place.
The White House in a statement later on Monday said that the two leaders discussed a range of topics in addition to the vote.
Earlier Monday, the White House had sidestepped questions on the referendum, with spokesman Sean Spicer saying the USA wanted to let an global commission review the results.
While some of the more far-reaching changes in Turkey would only come into effect after the next election - such as the power to issue decrees and appoint and dismiss ministers - others could into effect more quickly.
"Before we start getting into their governing system, let this commission get through its work", Spicer said.
Erdogan had a bumpy relationship with Trump's predecessor.
The referendum was widely seen as a plebiscite on Erdogan, who has led his country through more than a decade of economic growth and development, first as prime minister and then as president. The two men spoke frequently during the start of Obama's tenure.