The European Union also urged a probe into the poll fraud claims after international observers voiced concerns, but both US President Donald Trump and Russian President Vladimir Putin called Erdogan to offer congratulations.
Critics fear the changes will lead to autocratic one-man rule, but supporters say they simply put Turkey in line with France and the United States and are needed for efficient government.
The 'Yes' camp won Sunday's poll with just 51.41 percent of the vote but the result has been challenged, with angry protests staged in parts of Istanbul and other cities.
The changes, most of which are due to come into force after November 2019, are some of the most far-reaching in Turkey since Mustafa Kemal Ataturk established the modern state in 1923 on the ashes of the Ottoman Empire.
Bulent Tezcan, deputy leader of the main opposition Republican People's Party (CHP), formally requested that the Supreme Election Board (YSK) cancel the result.
The opposition is particularly incensed by a last-minute move by the YSK to accept ballot documents in envelopes without an official stamp.
"This is was a vote without legitimacy," Tezcan said.
Hundreds of people, some brandishing flares, marched through the anti-Erdogan Istanbul district of Besiktas in a new protest against the result, an AFP photographer said.
Meanwhile, in Kadikoy on the Asian side of the city, women banged drums and pans in a similar protest, with slogans like "We will not be scared, we will not be silent".
CHP chief Kemal Kilicdaroglu said the government and the YSK had "staged a coup against the national will".
The joint mission of the OSCE Office for Democratic Institutions and Human Rights (ODIHR) and the Parliamentary Assembly of the Council of Europe (PACE) said the YSK move on the stamps "removed an important safeguard".
They also said the campaign -- which saw the 'Yes' camp dominate the airwaves -- was conducted on an "unlevel playing field".
EU Commission spokesman Margaritis Schinas called on Turkish authorities "to launch transparent investigations into these alleged irregularities found by the observers".
In an interview with CNN, Erdogan denied claims Turkey was headed to dictatorship and that the new system was tailor-made for him.
The final results are due to be published in around 10 days and it is in this period that the YSK will consider the objections. Then the opposition can appeal to the constitutional court.
Keen to meet Trump
The new system will dispense with the prime minister's post and centralise the entire executive bureaucracy under the president, giving Erdogan the direct power to appoint ministers.
But the 'Yes' vote has even wider implications for Turkey, which joined NATO in 1952 and in the last half-century has been engaged in a stalled bid to join the European Union.
Erdogan reaffirmed he would now hold talks on reinstating capital punishment -- a move that would automatically end Turkey's EU bid -- and would call another referendum if the bill did not get enough votes in parliament to become law.
German Foreign Minister Sigmar Gabriel said that if Ankara were to bring back the death penalty, the move would be "synonymous with the end of (its) European dream".
In contrast with the EU's muted reaction, both Putin and Trump called Erdogan to congratulate him on his victory.
Erdogan told CNN he would be happy to have a face-to-face meeting with the new American leader "and take our relationship forward".
'Goal is to win'
In a blow to the president's prestige, the 'No' campaign notched up the most votes in Turkey's three biggest cities: Istanbul, Ankara, and Izmir.
Analysts have said the 'No' camp's performance was impressive especially given that the election was held under a state of emergency first imposed after July's failed coup.
But Erdogan, a one time semi-professional footballer, told CNN: "It does not matter if you win 1-0 or 5-0. The ultimate goal is to win the game."
Meanwhile, parliament agreed to extend the state of emergency -- already in place for nine months -- for another three months to July 19.
(This story has not been edited by NDTV staff and is auto-generated from a syndicated feed.)