Italy had insisted the Pinar was in Malta's search-and-rescue area, arguing that Malta should have accepted the migrants.
But Tonio Borg, Malta's foreign minister, told Al Jazeera that international law "provides that Malta co-ordinates the search-and-rescue and then the migrants go to the country closest to which they were received".
"They were three times closer to [the Italian port of] Lampedusa than to Malta, so I cannot understand why authorities are saying we should have accepted these migrants," he said.
Italy finally said it would allow the migrants to land for "humanitarian reasons".
Antonio Cacciatore, from the Porto Empedocle port authority, said the migrants would be identified and moved to immigrant shelters elsewhere in Sicily.
Twenty of the migrants suffering health problems, including a pregnant woman, were evacuated to Lampedusa at the weekend and the remaining 120 taken to Porto Empedocle in Sicily on Monday.
Tens of thousands of migrants pay smugglers to help them reach Italian shores each year, but their boats often capsize or get stranded.
About 36,900 illegal migrants arrived in Italy by boat in 2008, a rise of 75 per cent.
Malta, which has a population of around 350,000, has also seen an an increase in illegal migration, with about 500 migrants arriving since the beginning of the year.
It says its migrant arrivals form a greater proportion of its population than in any other country and are straining its social structure.