Some 40 co-ordinated rescue missions took place about 20km (12 miles) off the Libyan town of Sabratha, it added.
Video footage shows migrants, said to be from Eritrea and Somalia, cheering and some swimming to rescue vessels, while others carried babies aboard.
On Sunday more than 1,100 migrants were rescued in the same area.
The instability in Libya has made the country a hub for people-trafficking.
Monday’s operations involved vessels from Italy as well as the EU’s border agency Frontex and the NGOs Proactiva Open Arms and Medecins Sans Frontieres (MSF).
The migrants had set off in overcrowded and unseaworthy vessels with enough fuel to reach waiting rescuers, AP reported.
Last year more than one million migrants – many fleeing the civil war in Syria – arrived in Europe, sparking a crisis as countries struggled to cope with the influx, and creating division in the EU over how best to deal with resettling people.
In March, the EU struck a deal with Turkey to try to stop migrants crossing from Turkey to Greece while Balkan nations closed their borders to migrants. As a result, the number of arrivals using the so-called eastern Mediterranean route has fallen.
However, migrants from African countries such as Eritrea and Somalia as well as west African nations such as Nigeria and the Gambia are continuing to attempt the crossing from Libya to Italy.
Some migrants are seeking economic opportunities in Europe – others are fleeing war, instability or authoritarian governments.
About 106,000 people have arrived in Italy so far this year while 2,726 have died in the attempt, according to the International Organization for Migration (IOM).
The IOM says there are a further 275,000 migrants in Libya waiting to travel.
Overall, about 284,000 migrants have entered Europe so far this year through various transit routes across Africa, Asia or the Middle East.
Several vessels run by humanitarian organisations help patrol the route but risks were highlighted earlier this month when MSF said one of its boats was fired on by armed men.