Conte, whose government includes the anti-establishment 5-Star movement and far-right League, had earlier refused to endorse a summit text on security and trade until other leaders had pledged to help Italy manage Mediterranean arrivals.
European leaders also call on the European Commission to "swiftly explore the concept of regional disembarkation platforms, in close cooperation with relevant third countries".
The EU wants countries such as Algeria, Egypt, Morocco, Niger and Tunisia to process the claims of migrants seeking to travel to Europe with the aim of stopping many from making the final crossings.
Italy also wanted an end to the Dublin rules which dictate asylum-seekers must make their request in the first European Union state they set foot in.
Under the understanding with Greece and Spain, the two countries would be "ready to take back asylum seekers who in the future are identified by German authorities at the German-Austrian border" and who already have fingerprints in the EU's asylum database.
"We are satisfied", said Italian Prime Minister Giuseppe Conte, who at one point in the night threatened to veto any agreement unless more support was given to Italy. In Europe's case, the proposal also broadens an existing strategy of using African countries as partners that are willing to crack down on smugglers and intercept migrant vessels.
"The only thing European states appear to have agreed on is to block people at the doorstep of Europe regardless of how vulnerable they are, or what horrors they are escaping - and to demonise non-governmental search and rescue operations", MSF's emergencies chief Karline Kleijer said.
United Nations agencies report about 40,000 refugees and migrants have arrived in Europe by sea this year, a small fraction of the number over the same period in 2016.
"For now, the first-line countries are meeting their responsibilities", said Prime Minister Charles Michel, referring to European Union asylum rules which say that migrants must be dealt with by the first country in which they arrive.More news: Belgium beats England to top spot with Januzaj strike
More news: Republicans aim to debate immigration bill next week, ignoring Trump plea
More news: England fans flock to Russian Federation for group decider against Belgium
Donald Tusk, the European council president, told reporters in Brussels that it was "far too early to talk about a success".
France will not build migrant control centers on its territory given that, unlike Greece Italy and Spain, the country was not on the front-line of the migrant crisis, French President Emmanuel Macron said on Friday.
Migrants deemed eligible for global protection would be distributed among member states that voluntarily offer to take them in.
"In these troubled times, we need to present a face of unity, of efficiency in line with our values and shared histories", he said at a joint press conference in Paris with the Czech and Slovak prime ministers.
UNHCR spokesman Charlie Yaxley said the refugee agency is "still awaiting the legal analysis" of the new plan but would certainly welcome greater European Union collaboration on handling asylum claims.
"We took the view that we needed to have a degree of burden-sharing and the transfer of migrants to countries. The EU will continue to stand by Italy and other frontline Member States in this respect", says the resolution.
IOM's spokesperson Leonard Doyle said that "the majority" of these "should be in Europe", although these could be "potentially elsewhere", the agency said in a statement.
But the leaders failed to agree on long-stalled plans to overhaul the bloc's asylum rules, which say that migrants must be dealt with by the first country in which they arrive.