Despite the low turnout, the result was a "success for democracy and for a European Macedonia", Zaev said after the polls closed but before most votes had been tallied.
Macedonia's PM Zoran Zaev, his wife Zorica and his son Dushko cast their ballot for the referendum in Macedonia on changing the country's name that would open the way for it to join North Atlantic Treaty Organisation and the European Union in Strumica, Macedonia September 30, 2018.
Under a deal signed with Greece this summer, Macedonia agreed to change its name to the Republic of North Macedonia, while Greece agreed to lift its long-standing veto on Macedonia's NATO and European Union integration.
EU Enlargement Commissioner Johannes Hahn tweeted that he expects "all political leaders to respect this decision and take it forward with utmost responsibility and unity across party lines".
North Atlantic Treaty Organisation head Jens Stoltenberg said on Twitter the result was a "historic opportunity", adding: "NATO's door is open".
The polls closed with a turnout of almost 37% and the preliminary results showed people overwhelmingly supported the name deal with Greece and the Euro-Atlantic integration of Macedonia, according to data published by the election commission.
"MPs now have an obligation to make Macedonia a better place for all of us", Zaev told reporters late on Sunday.
"This is the will of the people and VMRO-DPMNE will follow it", party leader Hristijan Mickoski told reporters, adding that the "government lost its legitimity".
However, the party that ruled Macedonia for a decade until 2017 was split during the referendum campaign.
A poll published last Monday by Macedonia's Institute for Policy Research (IPIS) said between 30 and 43 percent of voters would take part in the referendum - below the required turnout.
Zaev said he would seek to secure the required two-thirds majority of the 120-seat parliament by next week for the constitutional changes.
For decades Greece and the former Yugoslav Republic of Macedonia (FYROM) had a contentious relationship over the name.
But the conflict goes back even deeper in history, with nationalists on both sides claiming that they were the true heirs of Alexander III of Macedon, also known as Alexander the Great, a fourth century BC warrior-king.
Zaev is billing the vote as a painful but historic opportunity to break the 27-year-old stalemate. According to Greece, the current name Republic of Macedonia implies territorial clams on the Greek northernmost province of the same name.
Supporters of a movement for voters to boycott the referendum hold a huge Macedonian flag as they celebrate in central Skopje after election officials gave low turnout figures.
Few Macedonians are enthused about the new name, saying they have been unfairly bullied by Greece.
Days before the voting, internet monitors at the Atlantic Council, a New York-based research group, said "Macedonia's information environment is becoming clouded with distorted and polarizing narratives by some Russian media outlets, especially from Sputnik - a state-funded online outlet".
But a desire to anchor their future to the West - and the economic prosperity that it could bring - has been a driving force behind the "yes" vote in one of Europe's poorest nations.
The referendum is Advisory, however, the majority of the members of the Macedonian Parliament has already declared that will support the results of the vote.
Europe and the United States have campaigned hard for the deal, with many leaders passing through Skopje this month to urge Macedonians to seize the "historic" opportunity.
At home, those in favour of the deal say a desire for an European Union future is helping bind ethnic Macedonians with the Albanian minority, who are broadly pro-West. Opponents to the deal have called for a boycott of Sunday's vote.