The main opposition party has launched an appeal to invalidate the result and some leading MEPs have called for Turkey's accession talks with the EU to be suspended.
The changes - due to be introduced before presidential and parliamentary elections in November 2019 - will turn Turkey into a presidential republic similar to the US and France. This could enable Erdoğan to stay in power until 2029.
EPP group Chair Manfred Weber said the referendum was "a historical break" and that Turkey was moving "away from the rule of law and democracy."
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"Turkish EU membership is off the table," he added.
S&D group leader Gianni Pittella agreed, saying the result was "another harsh blow against democracy and rule of law and another decisive step away from Europe."
The Italian MEP went on, "Erdoğan has closed his doors to the EU with this referendum. The EU accession talks should be suspended once Turkey decides to implement the 18 constitutional amendments.
"Many irregularities across the whole of Turkey have been alleged; irregularities that may cast a shadow on the final outcome. We, therefore, are looking forward to the official assessment by the OSCE/Office for Democratic Institutions and Human Rights (ODIHR) International Observation Mission.
"We remain convinced that the majority of the Turkish citizens are still looking at democracy and rule of law as crucial benchmarks. Unfortunately, Erdoğan is increasingly turning Turkey into a personal authoritarian regime.
"The S&D group will do whatever it takes to reinforce democracy and rule of law against the authoritarian drift carried out by Erdoğan. Nevertheless, the group requests the Council to envisage all the necessary measures in order to suspend the EU accession talks with Turkey. Europe cannot turn a blind eye and remain silent - and Europe will not."
ALDE group Chair Guy Verhofstadt said, "The proposed constitutional changes do not only grant President Erdoğan authoritarian powers, they also disrupt the democratic checks and balances in Turkey and the referendum on which they are based didn't meet the standards of the Council of Europe.
"A fair campaign has been prevented by the Turkish government who intimidated the 'No' side and even resorted to jailing dozens of journalists.
"On top of that, the outcome was very tight. Turkey's big cities, for example, clearly voted against changing the constitution.
"The only logical conclusion of this all is that the EU stops accession talks immediately and rethinks its relationship with Turkey."
Greens/EFA group co leader Ska Keller described the result as a "devastating blow to democracy" in Turkey.
The German deputy added, "After a campaign that took place in a climate of intimidation and fear, with outrageous restrictions on freedom of the press, Erdoğan is now on track to transform the country into a one-man authoritarian regime. The EU must now make clear that it stands on the side of those who, despite intimidation and prosecution, have stood up for democracy.
"It is now high time for the EU to increase the pressure on Erdoğan to end his fight against the free press and the opposition."
Her Greens/EFA group co-leader Philippe Lamberts added that the key refugee deal was now "destined to failure."
The Belgian member added, "All must surely now agree that it cannot possibly continue. Rather than making deals with Erdogan, the EU must stand strong behind its commitment to democracy, press freedom and human rights.
"Despite this result, now is not the time to turn our back on Turkey. We must continue a critical dialogue if we are to hope to influence developments in the country for the better."
Elsewhere, GUE/NGL group leader Gabi Zimmer expressed "disappointment and serious concern" adding that, "Erdoğan has abused Turkish democracy by winning a majority of the electorate in a referendum to change the constitution for personal benefit.
"His strategy was to insult journalists, Kurds, the opposition, dissenters and even EU countries and to declare them enemies. This made him look like a strong man and close ranks behind him."
However, ECR group leader Syed Kamall was less critical, saying the Turkish people had "expressed their will."
He said, "Whatever our views on President Erdoğan being granted more executive power, Turkey remains a Nato member and a key strategic partner in addressing common challenges such as the threat of terrorism, stability in the Middle East as well as the refugee and migration crisis.
"But this cooperation should not prevent us from being honest with Turkey, and with President Erdoğan. We will continue to express our concerns over the Turkish government's respect for liberal democracy, the rule of law and freedom of speech."