Italian Prime Minister Matteo Renzi had stalled on signing off on the deal because of questions about how the accord would work, but EU sources told AFP that Rome had given its approval on Wednesday.
Under the deal the European Commission, the EU executive, will contribute €1 billion to Turkey while the bloc's 28 member states will contribute two billion, the commission announced.
Germany is the country making the biggest contribution to the fund with €427 million, followed by Britain with €327 million and France with €309 million, EU sources said.
"I welcome the agreement by the Member States on the details of the refugee facility for Turkey," European Commission Vice President Frans Timmermans said in a statement.
"The money we are putting on the table will directly benefit Syrian refugees in Turkey, helping to improve their access to education and healthcare in particular.
"I also welcome the measures already taken by the Turkish authorities to give Syrian refugees access to the labour market and to reduce the flows."
Turkey - the main launching point for the one million refugees and migrants who arrived in Europe last year - has promised to cut the flow of people as part of the deal agreed with the EU at a summit in November.
The deal comes four days after Renzi and German Chancellor Angela Merkel held talks to break the logjam on the deal with Turkey.
Italy had questioned how much of the money should come from the EU budget, and how much control the bloc will have over how Ankara spends the funds.
Renzi has also traded barbs with Brussels about claims that Italy has been slow to set up so-called "hotspot" centres for registering and taking photos and fingerprints of newly arrived migrants.
But in a diplomatic show of unity on Friday after Renzi repeatedly criticised "German dominance" in EU affairs, both leaders said that the migrant crisis can only be solved if the 28 members of the bloc work together.
Across Europe, debate has raged on how to handle the biggest wave of migrants and refugees since World War II, many of them fleeing the war in Syria.
In January alone 360 people drowned while making the risky crossing of the Mediterranean, while more than 3,700 died in 2015, according to the International Organization for Migration.
Greece, where most of the refugees and migrants arrive, is under pressure from Brussels to improve its control of the EU's external borders, although Athens says there is little more it can do.
Italy and Greece were the main beneficiaries of an EU deal last year to redistribute 160,000 refugees around the bloc and ease the burden of frontline states, but only just over 400 have so far been relocated.