Earlier, Greece's right-wing defence minister resigned in protest at a deal ending a long running dispute with Macedonia over its name, saying he was taking his other six ministers in cabinet with him.
Parliamentary speaker Talad Xhaferi said 81 MPs voted in favour of the name change in the 120-seat chamber, securing the required two-thirds majority.
Greece has a parliamentary election scheduled for October and Syriza is now trailing the centre-right New Democracy party, which also opposes the deal with FYR Macedonia, by 12 points.
The two countries struck the deal on the new name in June previous year, but Macedonia will start using it only after the parliament in Athens also approves the change.
As promised, the country's main opposition party, VMRO-DPMNE, boycotted the vote, although a number of its now expelled members broke ranks to support the plans of Prime Minister Zoran Zaev's government to take the country into North Atlantic Treaty Organisation and the European Union, assuming Greek objections are ultimately lifted and Athens ratifies the deal.
In response to Kammenos' announcement, Tsipras said that he would request a vote of confidence in his government during the coming week. For years, Athens has blocked Macedonia's entry to the European Union and North Atlantic Treaty Organisation, citing the name dispute.
He said his party is quitting the government.
Kammenos has called the deal a national sell-out and had repeatedly threatened to leave if it came before the Greek parliament for ratification. He also said he requested Tsipras to take the Prespes deal to the Greek people in a referendum, a proposal which got nowhere.
"Within 10 days... if we see everything is in order, we will vote", Tsipras said on Friday evening.
Greece has a province called Macedonia and long demanded Skopje change its country name to remove what Athens considered to be an implied claim to Greek sovereign territory.
Zaev said on January 12 that the lawmakers had "made history", adding: "I know how hard that was".
Macedonia was founded in 1991 and recognized by the United Nations in 1993, but was accepted to the United Nations membership with the name of Former Yugoslav Republic of Macedonia (FYROM) due to Greek objection to the name Macedonia.
Zaev said he was "convinced that Greek lawmakers will recognise the historical significance of the agreement".
The ancient kingdom of Macedonia - or Macedon - was a relatively small part of the present day Greek province of Macedonia.
The naming row between the two countries began 27 years ago when FYROM declared independence from the Socialist Federal Republic of Yugoslavia, but has roots going back to antiquity.
Last June, Zaev and Tsipras reached a landmark compromise over the name dispute.