Italian Prime Minister Giuseppe Conte said five European countries had agreed to take in 250 of the 450 migrants aboard two EU border agency vessels.
Earlier on Sunday, Germany agreed with Italy to take in 50 migrants after Rome requested that its EU peers take some of the migrants stranded off the Italian coast.
A German government spokeswoman said in a statement that "Germany and Italy have agreed that, in view of the ongoing talks on intensified bilateral cooperation on asylum policy, Germany is ready to take in 50 people".
Conte had contacted his 27 EU peers to remind them that they had agreed at their end-of-June summit on the need to share the migration burden.
He announced on Saturday that France and Malta would take 50 migrants each, and "other countries will follow very quickly".
"This is the solidarity and responsibility we have always asked from Europe and now, after the results obtained at the last European Council, it is starting to become reality," he said on Facebook on Sunday.
"Let's continue on this path with firmness and respect for human rights."
Italy's new populist government, which came to power on June 1, wants to block any further migrant arrivals by boat and has banned NGO rescue ships from docking in Italian ports, accusing them of aiding human traffickers.
The latest migrants, like thousands of others, had set sail from Libya in a single wooden vessel which was identified early Friday.
Sunday morning, as the two Frontex vessels approached the boat, several migrants threw themselves overboard, prompting immediate efforts to rescue them, Italian sources said.
Eight women and children were taken to the Italian island of Lampedusa for medical treatment.
Spain's new foreign minister Josep Borrell meanwhile on Sunday said that the EU's prized Schengen free-movement system was "beginning to disappear" under pressure from migrants arriving in the bloc.
"Through the back door, France, Italy and Germany have placed controls on the borders because of the migration crisis," Borrell told the El Pais newspaper.
Spanish rescuers separately saved more than 340 migrants from the Mediterranean on Saturday.