Mr Tsipras earlier Sunday accepted the resignation of Defence Minister Panos Kammenos, who is head of his main coalition ally in government, the nationalist Independent Greeks party (ANEL).
Tsipras and Kammenos met for almost an hour before the prime minister said he had accepted Kammenos' resignation, but added that he would call for a confidence vote in parliament to keep his government alive.
In a parliamentary session on January 11, 81 Macedonian deputies in the 120-seat parliament voted in favor of the proposed changes to the country's constitution, 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. He also said he requested Tsipras to take the Prespes deal to the Greek people in a referendum, a proposal which got nowhere.
"We will proceed immediately to the renewal of the confidence in our government by the parliament in order to proceed with the major issues for our country", he said. Kammenos has already stated that the Independent Greeks would not support the ruling Syriza party.
Talking to reporters after their meeting, Tsipras thanked Kammenos for his "irreplacable and significant" contribution to the coalition government, and said that their partnership was frank and fruitful.
Macedonian lawmakers approved the agreement late Friday.
"Our parliament found the strength but it wasn't easy".
US Secretary of State Mike Pompeo welcomed the parliament's decision, praising the Macedonian leader's "vision, courage and persistence" in seeking a resolution to the dispute with Greece that would allow it to join North Atlantic Treaty Organisation and the European Union. "But I am convinced that the Greek parliament will also find the strength to make the decision", he said.
The wrangle had long frustrated Macedonia's European Union and North Atlantic Treaty Organisation aspirations.
But a change of government in Macedonia in 2017 finally led to the deal reached previous year.
Athens has for decades objected to the Former Yugoslav Republic of Macedonia's (FYROM) use of the word "Macedonia", arguing it suggests Skopje has territorial claims on the Greek border region of Macedonia.
Zaev said he was "convinced that Greek lawmakers will recognise the historical significance of the agreement".
For the Greeks, Macedonia is the name of a history-rich northern province that was the cradle of Alexander the Great's ancient empire. It began almost three decades ago, with Macedonia's declaration of independence, but has roots dating back centuries.
Last June, Zaev and Tsipras reached a landmark compromise over the name dispute.