Under the agreement, after Skopje cements the name change with constitutional amendments, Tsipras is to push the ratification of the agreement through the Greek parliament.
The opposition VMRO-DPMNE party has asked for an early election.
The Former Yugoslav Republic of Macedonia (FYROM) completed its side of the June agreement on Friday after it approved a change in the country's name to North Macedonia along with several amendments to the country's constitution.
Greece blocked its neighbour's aspirations to European Union and North Atlantic Treaty Organisation membership over the use of Macedonia which, it said, implied territorial claims to a Greek province of the same name.
But the Macedonian constitution stipulates that as the measure was passed by a two-thirds majority, the president has no choice but to sign it into law.More news: Repeated fast radio bursts from space make ‘alien life’ theories alive
The vote followed intense negotiations between Zaev's center-left coalition and some opposition lawmakers, who had initially agreed to back the agreement but raised last-minute objections.
The agreement on changing the name comes after a 27-year dispute, with Greece complaining that the name Macedonia implies claims on Greece's own territory and cultural heritage.
"I changed my opinion on the name issue in the name of progress and at the cost of my political career", he added.
Eighty-one deputies in the 120-seat parliament voted in favor, securing the required two-thirds majority, Radio Free Europe / Radio Liberty (RFE/RL) reported.
Separately, Greek Prime Minister Alexis Tsipras previously stated that they were planning to take this agreement to the Greek Parliament in January.More news: Ottawa Bus Crash: Westboro Station Incident Sees Double-Decker Vehicle Strike Transit Shelter
Tsipras spoke with Zaev on Friday to congratulate him after the vote, the Greek prime minister's office said.
The chief of NATO and the European Union's foreign policy chief have congratulated Macedonia for approving constitutional changes to rename itself North Macedonia under a deal with Greece.
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.
Macedonia approved the deal on Friday, while Greece's parliament is expected to vote on it before March.
Further complicating matters, Greek coalition partner Defence Minister Panos Kammenos, the leader of a smaller party, opposes the deal and has threatened to resign.More news: Megyn Kelly, NBC News reach separation agreement