However, key details remained to be worked out and the 28 leaders ordered more work by officials with a view to reaching an ambitious package deal with Turkey at their next scheduled summit, on March 17-18.
German Chancellor Angela Merkel and British Prime Minister David Cameron among others hailed the surprise Turkish proposal at an emergency summit in Brussels as a potential breakthrough in Europe's politically toxic migration crisis.
More than a million people fleeing war and poverty in the Middle East and beyond have flooded into the EU since early 2015, most making the perilous sea crossing from Turkey to Greece, then heading north through the Balkans to Germany.
"With this game-changing position in fact our objective is to discourage illegal migration, to prevent human smugglers, to help people who want to come to Europe through encouraging legal migration in a disciplined and regular manner," he told a news conference after the summit.
In exchange for stopping the influx, he demanded doubling EU funding through 2018 to help Syrian refugees stay in Turkey and a commitment to take in one Syrian refugee directly from Turkey for each one returned from Greece's Aegean islands, according to a document seen by Reuters.
He also asked to bring forward EU visa liberalization for Turks to June from end-2016 and to open five more negotiating chapters in Turkey's long-stalled EU accession process.
The EU leaders agreed to the earlier target date for visa-free travel provided Ankara meets all the conditions including changing its visa policy towards Islamic states and introducing harder-to-fake biometric passports.
They left open how much additional aid they would provide for refugees in Turkey and made only a vague reference to preparing for a decision on opening more areas of membership talks - a particularly sensitive issue for Cyprus.
