The debate on January 11 followed a postponement a day earlier, with Prime Minister Zoran Zaev and his allies negotiating with lawmakers opposed to the change.
Albanian Foreign Minister Ditmir Bushati tweeted after the vote, saying "the contribution of Albanian political parties once again proved to be a decisive factor".
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 has blocked the path to those global organisations since Macedonia broke away from the former Yugoslavia in 1991 because, it had said, the name Macedonia should apply exclusively to its own northern province.
Macedonia's parliament approved constitutional changes renaming the country North Macedonia, under a deal with neighboring Greece that will clear the way for Macedonia to join NATO and potentially the European Union.More news: 'Deadliest Catch' crab boat capsizes, kills three crew members, reports say
Macedonia's government was holding emergency talks for a second day on January 11 to try to secure a two-thirds parliamentary majority needed to pass a constitutional amendment that would rename the country the Republic of Northern Macedonia.
Ethnic Albanians make up about a quarter of Macedonia's 2.1 million people.
The opposition figure said the approval of the constitutional amendments, backed by 81 Macedonian lawmakers, was not a decision made by the people, adding it harmed the nation.
Zaev agreed on the name change with Greek counterpart and fellow leftist Alexis Tsipras in June.
"The prime minister congratulated Mr Zaev on the successful conclusion of the process to revise the constitution of the former Yugoslav republic of Macedonia", Tsipras' office said in a statement, after the vote. VMRO says the deal concedes too much to Greece.More news: Knicks and Kings reportedly talking Enes Kanter-Zach Randolph trade
Greece has previously blocked its neighbour's accession to North Atlantic Treaty Organisation and the European Union ever since Macedonia broke away from the former Yugoslavia in 1991.
The change is the result of a dispute between Macedonia and Greece over history and national identity that has lasted 27 years.
The Friday vote brought an end to months of political bickering in Macedonia that included a controversial consultative referendum in September and a long parliamentary battle.
However several opposition MPs broke ranks and voted for the change, alongside the ruling Social Democratic party and their junior coalition partners from the ethnic Albanian minority.More news: GM's Cadillac will introduce EV in fight against Tesla