Speaking after a phone call with Greek counterpart Alexis Tsipras, he said: "There is no way back".
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.
He added that Macedonia "cannot and will not be able in the future to claim any connection with the ancient Greek civilisation of Macedonia".
Greece had also vetoed its neighbor's bid to join North Atlantic Treaty Organisation and the EU.
"By solving the name question, we are becoming a member of NATO", Zaev added.
"The chance is here and it must be seized, bravely, as this is the only patriotic way", Macedonian Prime Minisrter Zoran Zaev told a press conference in Skopje.
After a long and thorough diplomatic process, both sides have come to an agreement to the name Republic of North Macedonia or Severna Makedonija, in Macedonian.
It was only with the adoption of a provisional name - the Former Yugoslav Republic of Macedonia (FYROM) - that the country was finally admitted into the United Nations in 1993.
Greek Prime Minister Alexis Tsipras. Credit PA
Both governments have faced criticism ahead of the compromise and on Tuesday Macedonia's President Gjorge Ivanov signalled his concern.
NATO and European Union officials welcomed the breakthrough, which NATO Secretary General Jens Stoltenberg said would help consolidate regional peace and stability.
Macedonia became a candidate for membership of the European Union in 2005 but Greece blocked the start of negotiations. However, Tsipras said, this will be contingent on Macedonia completing the constitutional changes.
Greek Defence Minister Panos Kammenos, whose right-wing Independent Greeks party is Tsipras' governing coalition partner, said he would oppose an agreement in a parliamentary vote, meaning the left-wing prime minister will need to seek support from political opponents.
In a statement, Nimetz congratulated the two prime ministers on reaching an agreement: "I have no doubt this agreement will lead to a period of enhanced relations between the two neighbouring countries and especially between their people".
Greece also demanded that Macedonia drops from its flag the rayed sun of Vergina, which it claimed was an ancient Greek symbol, as well as certain articles from its constitution.
Greek opponents of the deal say it would not go far enough.
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