They also said they would start work on possible debt relief for Athens, despite reservations on the part of powerful Germany which has pushed a more austere line.
The new 6.7-billion-euro tranche agreed by ministers in Brussels is the latest from Greece's third financial rescue package since 2010, when its debt crisis brought the euro close to collapse.
The current programme agreed in 2015 runs until August this year, after which the southern European nation hopes to fully return to market financing and get back on its own two feet.
EU Economic Affairs Commissioner Pierre Moscovici said 2018 "will be a decisive year for Greece".
"This will be the year when Greece finally leaves this long period of financial assistance, marked by very hard tests for the Greek people, but which allow Greece to emerge stronger and more resilient," France's Moscovici told a news conference.
Portugal's Mario Centeno -- chairing his first meeting of the Eurogroup of 19 finance ministers from the single currency -- said they would also start "technical work" on "debt relief measures" for Athens.
Greece's huge debt pile is equivalent to an unsustainable 180 percent of its annual economic output.
'Things have turned around'
Greek finance minister Euclid Tsakalotos -- a key figure in the 2015 bailout negotiations that nearly saw Greece crash out of the euro -- said it was a "very good meeting" for his country.
"People are now convinced that things have turned around, and people are beginning to talk about the future and Greece's exit from the programme," he told reporters.
The mention of debt relief was "particularly significant"," he added.
The latest tranche will be split into 5.7 billion euros paid in February and the remaining one billion paid later in the spring once eurozone officials have checked that Greece has carried out all the reforms, the Eurogroup said in a statement.
Centeno, chairing his first Eurogroup for the first time after replacing Jeroen Dijssebloem of the Netherlands, said the cash would cover debt servicing, arrears and boosting Greece's cash reserves ahead of the end of the bailout, said Centeno.
"This is critical to ensure Greece's full market access," he added.
Thousands of people demonstrated in Athens one week ago against the set of around 100 austerity measures imposed by Greece's creditors, which include a politically-charged curb on industrial action.
The reforms also allow for the foreclosure and auction of properties owned by bankrupted borrowers. Both measures were fiercely opposed by leftists and trade unions.
The Greek government insists that the changes are limited, and Prime Minister Alexis Tsipras rejected criticism "as a shameless lie" that his left-wing administration was out to make strikes illegal.
Debt-laden Greece has received three multi-billion-euro bailouts since 2010.
The current rescue programme -- a package worth 86 billion euros agreed after months of talks that almost saw Greece crash out of the euro -- is financially supported by eurozone states but not the International Monetary Fund.
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