Greece has agreed a deal to unlock a further 10.3bn euros in loans from its international creditors, after talks in Brussels.
Eurozone finance ministers also agreed to potentially offer Greece debt relief - a concession that was necessary to keep IMF on board.
The 19 ministers said the deal had been made possible by Greece's economic reforms and called it a breakthrough.
Greece needed this tranche of cash to meet debt repayments due in July.
Offering debt relief to Greece had been a key demand from the International Monetary Fund (IMF), which has been at odds with the Eurogroup (the 19 eurozone finance ministers) for months over the issue.
Germany in particular has been opposed to the idea of reducing debt - a so-called "haircut".
"We achieved a major breakthrough on Greece which enables us to enter a new phase in the Greek financial assistance programme," Eurogroup President Jeroen Dijsselbloem told reporters early on Wednesday.
He said a package of debt measures would be "phased in progressively".
This review was the first under Greece's third eurozone bailout, secured in August last year, after which Prime Minister Alexis Tsipras called a snap election.
The move came two days after the Greek parliament approved another round of spending cuts and tax increases demanded by its international creditors.
The bill also created a state privatisation fund requested by eurozone finance ministers.
Opponents of the measures demonstrated outside Parliament on Sunday.
The Greek government, led by the leftist Syriza coalition, agreed to a third bailout worth €86bn last year.
The Washington-based IMF will still have to give final approval at board level to the fund's continued participation in the Greek bailout.
The organisation says Greek public debt is unsustainable at current levels of about 180 percent of Greece's gross domestic product.
Spokesman Poul M Thomsen said: "We welcome that it is recognised that Greece needs debt relief to make that debt sustainable and it can't do it on its own."