On Sunday, Italian Prime Minister Giuseppe Conte said that five European countries had agreed to take in 250 people, with Spain, Portugal, France, Germany and Malta all offering to take in 50 of the group.
Rome had called on its EU peers to take in some of the migrants, following a June summit in which the countries agreed to share the migration burden.
Despite the controversy over who would take the migrants, Conte had earlier authorized the disembarkation of all 43 women and 14 children on board, who left the ships in the Sicilian port of Pollazzo.
Now all 450 are reported to have left the boats, and authorities are working to organize their travel to the other countries.
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"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," Conte said on Facebook on Sunday, referring to the distribution of the rescued people. "Let's continue on this path with firmness and respect for human rights."
Italy's new populist government, which came to power on June 1st, 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.
Matteo Salvini, Italy's hardline Interior Minister and head of the nationalist League party, called the result a "political victory" on Sunday.
He had earlier shared on social media the news of France and Malta's agreement to take in some of the migrants, using the hashtag "#primagliitaliani' (Italians first) and later wrote: "The next objective, to suppress once and for all the mafia of human trafficking, will be to take migrants back to where they left from."
Migrants en route to Europe are often anxious not to be returned to Libya, where they face rape and abuse in detention centres, and NGOs have criticized calls to simply return those who are rescued at sea. However, the rescuers face the dilemma of how best to help people without playing into the hands of people smugglers.
The latest migrants, like thousands of others, had set sail from Libya in a single wooden vessel which was identified early on Friday. On Saturday 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 said on Sunday 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.