Rescuers managed to save 144 of the migrants and were still searching for others after their vessel overturned 128 kilometres off the Libyan coast.
The total number of passengers on the boat and their nationalities was not yet known.
On Sunday alone the Italian coastguard rescued a total of 3,791 migrants, bringing the total to 5,629 since Friday, authorities said.
Recent good weather in the Mediterranean has prompted a surge in the number of migrants setting off for Italy aboard boats. As a result at least 10 rescue operations were underway on Monday.
A coastguard member said the 5,629 migrants rescued was not a record, but the number was very high for the month of April.
According to Italian authorities more than 15,000 migrants have arrived in Italy so far in 2015. There were 15,000 in April alone last year and an average of 25,000 each month between June and September.
Tunisia's coastguard and navy on Monday also rescued 174 migrants from two boats which broke down as they headed from Libya to Italy.
The Red Crescent said the migrants – 90 on one boat and 84 on the other – sent out a distress call from off the coast of southeast Tunisia as they headed for the Italian island of Lampedusa.
"It's clear that the flux of migration from Libya is going to continue," said William Lacy Swing, director general of the International Office of Migration (IOM).
The head of the European Union's border control agency Frontex said last month he fears as many as a million migrants could try to reach Europe this year from Libya alone.
The number of migrants entering the European Union illegally in 2014 almost tripled to 276,000, according to Frontex, nearly 220,000 of them arriving via the often dangerous Mediterranean crossing.
An increasingly violent and chaotic situation in Libya, a key jumping off point for migrants, has also helped prompt the huge hike in the number of asylum seekers trying to reach Europe.
About 400 people, according to the IOM, have died in recent months, fuelling criticism of rescue efforts.
The UN's refugee agency UNHCR in February slammed as "woefully inadequate" Triton, the Frontex-run maritime border patrol which, since November, has replaced Mare Nostrum, a much bigger search and rescue operation that was run by the Italian navy.
Italy decided to scale back the mission after its EU partners refused to share running costs of around €9 million ($10 million) a month.
Triton has a monthly budget of €2.9 million and its patrols are generally restricted to the territorial waters of EU member states.
People smuggling remains a lucrative business for traffickers, with migrants rescued in the Mediterranean in February saying they paid between $500-$1,000 for their crossing in boats that are often no more than rubber dinghies.