The agreement unlocks 10.3 billion euros (USD 12 billion) in bailout cash that Greece needs to repay big loans to the European Central Bank (ECB) and International Monetary Fund (IMF) in July, having already fallen behind in paying for everyday government payments and wages.
The US-based IMF has made easing Greece's huge debt burden a condition for its continued participation in the bailout programme, despite opposition from Germany to giving Athens more favours.
Eurogroup chief and Dutch Finance Minister Jeroen Dijsselbloem called the deal, hammered out at late-night talks in Brussels, a "major breakthrough".
The bailout rewards Athens for meeting the terms of its 86-billion-euro bailout programme agreed last July.
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Greece's creditors will pay a first 7.5-billion tranche in June and the rest in a series of later disbursements.
Greek government bonds soared in response to the deal, pushing the yield on the benchmark 10-year bond to below 7 per cent for the first time since November as the once very real chance of Greece quitting the eurozone was now seen as remote.
The Athens stock market firmed initially, but quickly turned lower, losing 0.3 per cent in mid-afternoon.
The hardest part of the talks was defusing the row between Greece's creditors, the eurozone governments and the IMF, over the state of the Greek economy and debt relief.
"The Eurogroup agreed today on a package of debt measures which will be phased in progressively," said Dijsselbloem.
"For the first time there is a clear and binding roadmap for debt relief," Greek government spokeswoman Olga Gerovassili said.
"In a spectacularly sophisticated show of kicking the can down the road, debt relief will be considered - later," said Paul Donovan, an economist at UBS. "With nothing so vulgar as amounts of money being discussed.