Lawmakers in Athens tentatively planned to hold the confidence vote on Wednesday, months before regular elections are due in September.
Mr Tsipras's party has 145 seats in the 300-seat Parliament.
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.
He also said that his right-wing populist Independent Greeks (ANEL) party was quitting the government.
Prime Minister Tsipras still needs parliament to ratify the deal and that has become more hard.
Kommersant said that an unnamed European diplomatic source in the Balkans "confirmed that North Atlantic Treaty Organisation intends to sign a protocol on membership with Skopje in February, which will, after the ratification by all alliance members, become the 30th member state".More news: Kingdom Hearts 3 Post-Launch Plans Include Epilogue and 'Secret Movie'
The impact of the resignation on the governing coalition was not immediately clear.
Tsipras hopes to rely on lawmakers from the small pro-EU To Potami (The River) party to get the agreement approved, while some of ANEL's lawmakers also remain ambivalent to the deal.
Kammenos has called the deal a national sell-out and had repeatedly threatened to leave if it came before the Greek parliament for ratification.
Kammenos had said on Sunday that he would dismiss ANEL ministers who insisted on keeping their ministerial seats ahead of a vote of confidence in the government this week.
Prime Minister Alexis Tsipras said, in response, that he will ask for a vote of confidence in Parliament in the coming week.
ANEL is the junior party in Greece's ruling coalition and strongly opposes the name change, a rift that threatens to destabilise the government of Prime Minister Alexis Tsipras, who helped broker the deal.More news: White House asked for options to strike Iran
Tsipras has said a separate vote on a protocol to enable Macedonia to join North Atlantic Treaty Organisation would be held "not long afterwards".
Fearing Macedonia would try to annex Greek Macedonia, the Greece Government refused to accept the new country's name, and faced the United Nation to refer to it as "the former Yugoslav Republic of Macedonia".
"Our parliament found the strength but it wasn't easy".
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. 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. "But I am convinced that the Greek parliament will also find the strength to make the decision", Zaev told a press conference in Skopje.
And, he concluded, Macedonia is a name intrinsically linked with the Greek culture, which recalls the triumphs of the era of warrior king Alexander the Great, king of Macedonia.
The country's main opposition, the conservative New Democracy party, said it will block the deal.More news: Oil prices rise amid OPEC-led output cut last week