Greece and Macedonia reached a historic agreement Tuesday to end a bitter 27-year name dispute that had kept the smaller and younger country out of global institutions such as North Atlantic Treaty Organisation, the two countries' prime ministers announced.
"We have a deal", Mr Tsipras said.
"We have a deal, I'm happy because we have a good deal which covers all the preconditions set by the Greek side", he said.
Mr Tsipras said the deal met the concerns of Greece that a revised name for Macedonia be a "compound" name with a geographical qualifier.
There was no immediate comment from Macedonian officials.
"Maybe what has the most historic gravity and value for Greece (is that) according to this accord. our northern neighbours don't have, and can not assert, any link to the ancient Greek culture of Macedonia", Tsipras said in a televised address on Tuesday evening.
The name dispute has soured relations between the two neighbours at least since 1991, when Macedonia broke away from former Yugoslavia, declaring its independence under the name Republic of Macedonia.
Athens has long objected to its neighbour's constitutional name - the Republic of Macedonia - because it fears it could imply territorial ambitions.
In the 1995 interim bilateral accord, Greece agreed that the term "the Former Yugoslav Republic of Macedonia" (FYROM) would be applied to Macedonia until the dispute is settled.
"A deal could be announced even today", said an unnamed government official cited by Reuters.
The "new" Slavic people calling themselves Macedonians did not help matters when they named the main airport in the capital, Skopje, after Ancient Greek hero Alexander the Great, as well as a key highway running from Skopje to the Greek border.
And main opposition parties in both countries rejected the agreement.
Greece will then back invitations for Macedonia to join North Atlantic Treaty Organisation and start negotiations on joining the EU.
"In other words, if the constitutional amendment is not successfully completed, then the invitation to join North Atlantic Treaty Organisation will be automatically rescinded and the accession talks with the European Union will not start", he said. I believe in the democracy of the internet and want to keep this site and its enriching content free for everyone.
Greek Foreign Minister Nikos Kotzias had prepared a 20-page draft agreement after repeated talks with his Macedonian counterpart Nikola Dimitrov. "We have a combined name with a geographic qualifier for all uses", Tsipras said as quoted by the Athens-Macedonian News Agency (AMNA) news agency.
European Council President Donald Tusk tweeted his support for the agreement: "Thanks to you the impossible is becoming possible".
European Union foreign affairs chief Federica Mogherini and commissioner Johannes Hahn issued a joint statement congratulating the two prime ministers "in reaching this historic agreement between their countries, which contributes to the transformation of the entire region of southeast Europe".
However, both prime ministers faced dissent at home.
This year there have been several protests against an agreement in Skopje, as well as in Athens and Greece's second city of Thessaloniki in the north. First the Macedonian parliament and people have to vote on the proposed new name.