'There is an agreement. "There is no way back", Zaev told a news conference.
The United Nations has been calling it the Former Yugoslav Republic of Macedonia.
But Greek Defense Minister Panos Kammenos, whose right-wing Independent Greeks party is Tsipras' junior coalition partner, said he would reject the name agreement in a parliamentary vote.
"We have a good agreement that covers all the preconditions the Greek side had set", he said, adding that Macedonia would revise its Constitution for the name change and that the deal "secures the historic heritage of ancient Greek Macedonia".
In 2008, Macedonia presented itself as a candidate for membership of the North Atlantic Treaty Organisation military alliance under the provisional name but again faced a Greek veto.
European Council President Donald Tusk tweeted his support for the agreement: 'Thanks to you the impossible is becoming possible'.
"I am encouraged by the dedication of both governments to deliver mutual benefits for all their citizens through the establishment of a strategic partnership as a basis for intensified cooperation across all sectors", he said in a statement.
Mr Zaev said the deal would be signed this weekend, and a voter referendum would be held in the fall.
The name would be the Republic of North Macedonia, subject to approval.
Macedonia agrees to new name after 27-year dispute with Greece
According to reports this will most likely take place at Prespes in northern Greece on Saturday.
"Accepting the "Macedonian language" and the "Macedonian ethnicity" constitute a non-acceptable national concession", he said on his official Twitter account.
'It's clear that (Macedonia) bears no relation to the ancient (Macedonian) culture. and that their language belongs to the Slavic language, ' he said.
Both governments have faced criticism ahead of the compromise and on Tuesday Macedonia's President Gjorge Ivanov signalled his concern.
He is close to the nationalist VMRO-DPMNE party which was defeated by Zaev in elections a year ago.
"We have been solving a two-and-a-half-decade dispute. that has been drowning the country", he said, adding that the deal "will strengthen the Macedonian identity".
The tiny Balkan country has repeatedly tried to join the European Union and North Atlantic Treaty Organisation, but its attempts have been blocked by neighbouring Greece.
The EU's diplomatic chief Federica Mogherini said the deal "contributes to the transformation of the entire region of southeast Europe".
Athens officials had objected to the name Macedonia, fearing territorial claims on a northern Greek province of the same name.
Greece's Macedonia is a northern region that includes the country's second city, Thessaloniki, whereas Macedonia's Macedonia is, well, a country.