The future of the ruling coalition government remains unclear. However some of his ANEL party's MPs remain ambivalent.
Prime Minister Alexis Tsipras continues to declare his belief that the deal will be ratified with the support of a few centre-left and independent lawmakers, filling in for the loss of ANEL support.
Tsipras said he had a "frank discussion" with Kammenos, whom he thanked for his government partnership, and announced that Admiral Evangelos Apostolakis, chairman of the Joint Chiefs of Staff, will become the country's new defence minister.
Before the vote in the Macedonian capital Skopje, a rift seemed to be growing over the issue between Prime Minister Tsipras and his right-wing coalition partner, Panos Kammenos, who has threatened to leave the government, The NYT report said.
But Mr Kammenos and other nationalist forces including the main Greek conservative opposition party, New Democracy, said they can't sanction this name as they identify with north Macedonia a northern region of Greece.
"Our parliament found the strength but it wasn't easy".
Zaev, who came to power in May 2017, is now looking to his Greek counterpart Alexis Tsipras to uphold his end of the deal, which the pair brokered a year ago.
The Macedonian Assembly has backed changing the country's name in the Republic of North Macedonia, thus opening the way towards NATO and the EU.
Zaev said on January 12 that the lawmakers had "made history", adding: "I know how hard that was". In ancient times it was the cradle of Alexander the Great's empire, a source of intense pride for modern-day Greeks.
The new name will be used both internationally and bilaterally, so that even the 140 or more countries that recognise the name Macedonia will also have to adopt North Macedonia.
A spokesperson for the governing Social Democrats said the ethnic Albanian legislators also agreed to back the deal.