Celebrations and protests both occurred in the streets of the major cities, but so far, there was no sign of violence between the opposing parties.
Prime Minister Binali Yildirim, who led the Yes campaign alongside Mr Erdogan, said: "This referendum once again has proven the level of maturity and the improvement of Turkish democracy to the rest of the world".
Erdal Aksunger of the Republican People's Party called for a partial recount.
A narrowly won vote Sunday in Turkey to replace the current parliamentary system with a strong executive presidency fell short of a genuine democratic process, global observers said Monday.
German Chancellor Angela Merkel, in a joint statement with Foreign Minister Sigmar Gabriel, said "The (German) government expects that the Turkish government will now seek respectful dialogue with all political and social forces in the country, after this tough election campaign".
The referendum won Sunday - as confirmed by the head of Turkey's electoral board although the vote count will continue for more than a week - with "yes" votes leading "no" votes by 51.2 percent to 48.8 percent, with over 47 million votes counted, according to the state-run Anadolu Agency. That caused Erdogan to accuse German and Dutch officials of acting like Nazis which, in turn, prompted strong condemnation of the Turkish president's words from European leaders.
Merkel and Gabriel said that Turkey - as an OSCE member and European Union candidate country - needs to consider these concerns.
He has already been the most powerful man in Turkey for 14 years, and though he was democratically elected, he's a divisive politician, and his critics say the referendum is a step towards dictatorship, and it could change Turkey irrevocably.
The power of publishing decrees will shift from the cabinet to the president who will intervene in appointing judiciary members, and will hold executor powers instead of the prime minister and parliament.
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Erdogan, whose narrow victory laid bare the nation's divisions, told flag-waving supporters that foreign election observers should "know their place" and Turkey did not "see, hear or acknowledge" criticism that the vote did not live up to worldwide standards.
"Support for the opposition from a number of European capitals indicates that the West finds the Turkish president's growing powers worrisome", Slutsky said.
The country's pro-Kurdish party said it may take the case to the European Court of Human Rights if the electoral board does not reverse its decision and nullify the ballots lacking the official stamps.
Erdogan has long said the changes to the constitution were needed to end the chronic instability that plagued the country over decades when the military repeatedly tried to seize power from weak civilian governments.
Tezcan said it was not possible for authorities to determine how many ballot papers may have been irregularly cast.
They set a limit of two five-year terms for presidents and also allow the president to remain at the helm of a political party.
The new presidential system takes effect at the next election, now slated for 2019.
"Had Erdogan lost this vote, he would have lashed out at his enemies and become consumed with trying to bring the issue back to vote", Jeffrey said.
The outcome could likewise have even more extensive ramifications for Turkey which joined North Atlantic Treaty Organisation in 1952 and for the last half-century has set its sights on joining the European Union.
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