However, only a third of the 1.8 million-strong electorate voted.
The low turnout was anticipated and the authorities in Skopje had suggested they were prepared to accept as legitimate the referendum's result even with 40% participation.
Macedonian citizens on Sunday voted on a deal reached with Greece, while the referendum question was formulated this way: "Are you in favor of membership in the European Union and North Atlantic Treaty Organisation by accepting the agreement between the Republic of Macedonia and the Republic of Greece?"
Greece believes that the opportunity to solve a decades old dispute over the Macedonia name issue must not be wasted, the country's government spokesman said on Monday, a day after a referendum was held in the ex-Yugoslav Republic.
Officials in Zaev's government have said they have 71 parliamentary deputies ready to approve a constitutional amendment accepting the name change, short of the two-thirds majority, or 80 votes in parliament, needed to amend the constitution. "I now expect all political leaders to respect this decision and take it forward with utmost responsibility and unity across party lines, in the interest of the country". Under the country's constitution, a binding referendum would need a minimum turnout of 50 per cent.
His governing coalition partner, Panos Kamenos, head of the right-wing Independent Greeks, vows to vote against the agreement in the Greek Parliament.
"MPs now have an obligation to make Macedonia a better place for all of us", Zaev told reporters late on Sunday.
"The government has lost its legitimacy, and the only thing remaining is to respect the will of the people", Mr Hristijan Mickoski, leader of the main centre-right opposition party, VMRO-DPMNE, wrote on Facebook. Zaev said that if opposition parties did not support the constitutional changes, then he would resort to "another democratic tool" - a call for early elections as soon as November - to ensure that the changes would be made.
The referendum offered a chance for the Macedonian public to weigh in on the deal with Greece, which has kept Skopje from being a full participant in Western institutions since it declared independence from Yugoslavia in 1991.
At the time, Athens kicked up a major fuss, accusing Skopje of stealing the name of its own northern province, which is also called Macedonia.More news: Ohio State rally stuns Penn State
The dispute dives deep into history, with both countries vying for claims to Alexander the Great's ancient empire of Macedon, which spanned their territories.
Zaev is billing the vote as a painful but historic opportunity to break the 27-year-old stalemate.
An editorial in the same paper said: "It would not be an exaggeration to say that a large part of the Greek political system was relieved by the absenteeism in the referendum. because any failure to ratify the agreement will burden (Macedonia), not Greece".
Few Macedonians are enthused about the new name, saying they have been unfairly bullied by Greece.
Zaev, the prime minister, has said he would resign if the "No" vote prevails, and the deal with Greece would be dead.
Europe and the United States have campaigned hard for the deal, with many leaders passing through Skopje to urge Macedonians to accept the "historic" opportunity.
Russia, however, is not keen on North Atlantic Treaty Organisation expanding in a part of Europe once under its sphere of influence.
"Macedonia has spoken today - Macedonia said - the deal is off".
In the lead up to the referendum, German Chancellor Angela Merkel, Austrian Chancellor Sebastian Kurz, NATO leader Jens Stoltenberg, former US Vice President Joe Biden and former President George W. Bush were among some of the many foreign dignitaries who either visited or spoke in favour of a "yes" vote.
The Greek parliament must also give its approval for the change to go ahead.