"This historic agreement is testament to many years of patient diplomacy, and to the willingness of these two leaders to solve a dispute which has affected the region for too long", said NATO Secretary General Jens Stoltenberg in a statement.
Both governments have faced criticism ahead of the compromise and on Tuesday Macedonia's President Gjorge Ivanov signalled his concern. The new name will be used both domestically and internationally, with Macedonia making a relevant amendment to its Constitution, Zaev added.
By solving the dispute which has lingered for more than 25 years, Macedonia will remove the biggest obstacle on its path towards European Union and North Atlantic Treaty Organisation accession.
People's nationality will be defined as "Macedonian/Citizens of the Republic of North Macedonia".
Greece and Macedonia have reached an historic accord to resolve a dispute over the former Yugoslav republic's name that has troubled relations between the two neighbours for decades.
Albanian Prime Minister Edi Rama called his Macedonian and Greek counterparts "two progressive leaders" and used their Twitter handles to give their praise, tweeting: "Balkans are today a better place thnx to you both".
Athens objected to its neighbor's new name, saying it implied a territorial claim over Greece's province of the same name, which borders the Balkan country.
"We have a deal, I'm happy because we have a good deal which covers all the preconditions set by the Greek side", said Greek Prime Minister Alexis Tsipras.
The dispute over the "Macedonia" name had been a thorn in relations of the two countries at least since 1991, when Macedonia broke away from former Yugoslavia, declaring its independence under the name Republic of Macedonia.More news: Tesla cutting 9% of jobs - Elon Musk
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They also urged the Council to endorse opening EU accession talks with Macedonia, which the European Commission recommended in April.
"We have a deal", Mr Tsipras said."Thanks to you, the impossible is becoming possible", he said.
But before that, he said the agreement will be put to parliament for ratification in order not to waste any time and to allow Greece to lift its blockade on Macedonia's potential membership of North Atlantic Treaty Organisation and the EU.
European Union foreign affairs chief Federica Mogherini and Johannes Hahn, commissioner for European Union enlargement, said they looked forward to accession negotiations beginning with Skopje in June.
Greece had long demanded that Macedonia change or modify its name to avoid any claim to the territory and heritage of the region, the birthplace of Alexander the Great.
But Macedonia feared that a name change could have a negative impact on the national identity of its people, the majority of whom simply call themselves "Macedonians".
But Greek Defense Minister Panos Kammenos, whose right-wing Independent Greeks party is Tsipras' junior coalition partner, said he would reject the name agreement in a parliamentary vote.
Last week in Pella, Greece, protesters opposed the use of the name "Macedonia" in any solution to the dispute between Athens and Skopje.
A resolution of the dispute would see Greece lift objection to Macedonia's accession to North Atlantic Treaty Organisation.