However, the ball is now in Athens' court as the name change will be effective only once it is cleared by the Greek parliament.
That prompted VMRO-DPMNE leaders to denounce "bribes and threats" used by Zaev to reach the necessary majority.
"A better deal could not be reached, and without an agreement with Greece there will be no North Atlantic Treaty Organisation and European Union (membership)", Zaev said.
The change is the result of a dispute between Macedonia and Greece over history and national identity that has lasted 27 years.
Macedonia's rightwing president, Gjorge Ivanov, has been a vocal opponent of the proposed name change and continues to speak out against it.
"I changed my opinion on the name issue in the name of progress and at the cost of my political career", he added.
Prime Minister Zoran Zaev said the move would "open the doors to the future, Macedonia's European future".
German Foreign Minister Heiko Maas hailed the vote, expressing expectations "that Republic of North Macedonia will soon be able to join NATO and start discussions for entry into the European Union".
Zaev's governing coalition needed opposition backing to get the required number of votes and had said on January 10 that it was struggling to achieve that after a small ethnic Albanian party raised last-minute objections.
The vote in Athens threatens to destabilise Tsipras' ruling coalition, as the junior party in government, the nationalist Independent Greeks (ANEL) party, strongly opposes the Prespes agreement.
Tsipras has said he sees the Macedonia deal as "one of his greatest legacies" as premier, second only to leading Greece out of the bailout era.
Demonstrators wave flags in front of the parliament building in Skopje on 23 June 2018 during a protest against the new name of the country, the Republic of North Macedonia.
Tsipras has said he wants to bring the deal to Greece's parliament in coming weeks.
Macedonian lawmakers were convening later Friday to vote on the amendments, for which a super majority of two-thirds of the 120 members - or 80 votes - is required.
Greece is bound by terms of the deal to stop blocking Macedonia from NATOand other worldwide groups and to allow it to start European Union accession talks as part of the deal.