The bailout fund will disburse €8.5bn (£7.4bn) to Greece, eurozone ministers said in a statement.
The latest tranche of the international bailout will help avert a fresh debt crisis in July when the next €7bn euro repayment of loans becomes due.
The payment is still subject to parliamentary approvals in some countries.
International Monetary Fund (IMF) director Christine Lagarde said she would propose an approval in principle to her executive board.
The IMF wants clarity on longer-term debt relief for Greece once the current funding scheme, worth up to €86bn, runs out next year.
Ms Lagarde said the IMF was ready to participate to the third bailout programme for Greece after the meeting of eurozone finance ministers in Luxembourg which capped months of negotiations.
However, the IMF could join the programme with a financial support "in the range of $2bn" only after a full deal on additional measures of debt relief for Greece, she said.
Time was beginning to press for this payment. Greece has repayments on other loans due next month, which it could not otherwise have made.
The decision by eurozone finance ministers reflects economic policy actions already taken by Greece and the new commitment by the IMF's managing director Christine Lagarde to recommend that her board contribute financially to this bailout.
An IMF contribution was politically important for Germany, especially to strengthen the perceived credibility of the bailout.
Analysis: BBC economics correspondent Andrew Walker
Another laborious step in this very long (I'll resist any Greek clichés at this point) story.
We have got used to delays resulting from Greek politics - a referendum, elections, problems in parliament holding up economic policy reforms that Greece was required to make.
This time it was more about German politics and the insistence on an IMF financial contribution while being reluctant to agree up-front the commitments on debt relief that the IMF thinks essential to make the Greek finances sustainable.
The result is the IMF agreeing in principle to contribute provided the Eurozone do eventually make a commitment on debt.
It's untidy and Christine Lagarde called it second best. And it is not a large contribution that the IMF will make.
She suggested it would be about $2bn against a Eurozone commitment to provide more than forty times that amount.
The eurozone group welcomed the financial and structural reforms enacted by Greece, including income tax and pension reform.
"I am pleased to announce we have achieved an agreement on all elements," Eurogroup head Jeroen Dijsselbloem said.
Germany's parliament will discuss the bailout deal on Friday, finance minister Wolfgang Schaeuble said.
"Greece has to become competitive to get access to debt markets so it can stand on its own two feet," Mr Schaeuble said.
"For that Greece has to carry out reforms."
Greek finance minister Euclid Tsakalotos said the agreement would allow market access 'in due course'.
"There is now light at the end of the tunnel," Mr Tsakalotos said.
