The navy said in a statement that the San Giorgio, an amphibious transport dock ship, had picked up the highest number - 998 migrants including 214 women and 157 minors.
Four merchant ships flying the Hong Kong, Italian, Moldovan and Panamanian colours that were sailing in the area also assisted in the rescue operations, the navy said.
A navy frigate, the Bergamini, brought 443 of the migrants to shore on Thursday in the port of Augusta in Sicily. One of them was immediately arrested on charges of people smuggling for allegedly captaining the migrant boat.
Most of the asylum-seekers arriving in Italy come from Eritrea or Syria and are departing from Libya. Many also come from impoverished parts of sub-Saharan Africa.
Thousands have died in tragedies at sea in recent years because they often cross in rickety fishing boats that are badly overcrowded to maximise profits for traffickers.