"There is no way back," Zaev said at a news conference after speaking with his Greek counterpart, Alexis Tsipras, by telephone. A meeting of the two soon may seal the deal, he said.
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"Our bid in the compromise is a defined and precise name, the name that is honorable and geographically precise — Republic of Northern Macedonia."
"By solving the name question, we are becoming a member of NATO," Zaev added.
The accord still requires ratification by the two national parliaments and a referendum in Macedonia. It also needs to revise its constitution, Tsipras said, before Greece ratifies the deal.
The name dispute has soured relations between the two neighbors since at least 1991, when Macedonia broke away from the former Yugoslavia, declaring its independence under the name Republic of Macedonia.
Many Greeks believed that their northern neighbor was trying to hijack Greece's ancient cultural heritage. Macedonia the Greek province is the birthplace of Alexander the Great.
"Maybe what has the most historic gravity and value for Greece (is that) according to this accord ... our northern neighbors don't have, and cannot assert, any link to the ancient Greek culture of Macedonia," Tsipras said in a televised address on Tuesday evening.
But Greeks seem cool to any deal involving the continued use of the name Macedonia by their northern neighbor. Most opposition parties have criticized Tsipras's tactics, and even his coalition partner, the right-leaning Independent Greeks, have said they will not back an accord that allows the continued use of the name Macedonia.
But Tsipras is still likely to win support from center-left parties. "We want to be part of a solution," said an official at the opposition Socialist Party.
Macedonia and Greece have been racing to agree on the outline of a settlement before an EU summit in late June. A NATO summit is scheduled for mid-July.
Veteran United Nations diplomat Matthew Nimetz, who has been a mediator in the name dispute since 1994, hailed the "leadership, vision and determination" of the foreign ministers of Greece and Macedonia, who have negotiated for months.
"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.