The name change will only be effective once it is also agreed by the Greek parliament.
Athens and Skopje reached the agreement on renaming Macedonia in June.
The disagreement has sparked protests on both sides of the territorial boundary and produced real consequences: Greece has blocked Macedonia's entry into both North Atlantic Treaty Organisation and the European Union.
Macedonian deputies voted Friday to change the country's name to "the Republic of North Macedonia", settling a decades-long row with Greece and paving the way to NATO and European Union membership.
"Within ten days, in any case as soon as the (Macedonia parliamentary vote) result is notified to us and if we see that everything is in order, we will vote (to approve) the Prespes Agreement", he told Open TV.
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FYR Macedonia is controversial for Greeks because it uses the same name as a neighbouring region in northern Greece.
Parliamentary speaker Talad Xhaferi said 81 MPs had voted in favour of the name change in the 120-seat chamber, securing the required two-thirds majority.
Opponents to the proposal say they are defending Macedonia's name, identity, and history, as well as the traditions of the Macedonian people, against what they call "the greatest national treachery".
Delays had marked the October vote that launched the procedure to change the constitution, also with a two-thirds majority. In September, the move was supported by over 91 percent of Macedonians at a referendum. Representatives of the opposition VMRO-DPMNE, who opposed the agreement with Greece, boycotted the vote.
Greek Prime Minister Alexis Tsipras on Friday congratulated Zaev after the vote.
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And he's still threatening to declare a national emergency to circumvent Congress if he can't get the wall money from them. Senators like yourself, who are seeking to strip Americans of their Constitutional right to free speech".
The opposition VMRO-DPMNE party has asked for an early election.
"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.
Prime Minister Zoran Zaev's governing coalition needed opposition backing to get the required number of votes and had said Thursday it was struggling to achieve that after a small ethnic Albanian party raised last-minute objections.
Greek Prime Minister Tsipras has said he wants to bring the deal - which has brought his coalition government to the brink of breakup - to parliament in coming weeks.
Athens argues that use of the term "Macedonia" implies territorial claims on Greece's northern province of the same name and on its ancient heritage.
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