The Italian coast guard told Efe news that some 1,200 people arrived in the Sicilian port of Messina, and that sea conditions were rough.
But at least 20 drowned as aid workers struggled to rescue more than 1,500 migrants in one ongoing operation.
But MSF have defended its actions on Twitter, as they claim the same number of migrants would try to cross, even if they weren't there.
Doctors Without Borders (MSF) boats Prudence and Aquarius rescued some 1,145 people from nine dinghies in non-stop operations.
The Italian coastguard and other boats rescued some 3,000 migrants from unseaworthy boats off the Libyan coast on Saturday, as the numbers of those attempting to reach Europe from Africa rises amid good weather, a participating NGO said.
The Greek coastguard on Saturday discovered a boat in distress carrying around 40 migrants, including 13 children, between Greece and Italy.
More than 24,000 migrants have crossed the Mediterranean Sea from Libya to Italy over the first three months of 2017, many more than the 18,000 people who crossed during the same time period in 2016, according to the UN High Commissioner for Refugees (UNHCR).
According to the International Organization for Migration (IOM), over 32,000 refugees have arrived in Europe so far this year.
Hailing mainly from sub-Saharan countries, most of the migrants board boats operated by people traffickers in western Libya, and make for the Italian island of Lampedusa 300 kilometres (190 miles) away.
On Friday, rescue vessels saved more than 2,000 people from flimsy dinghies.
Sami said the weekend's work had a frenetic pace, to the point where some humanitarian agents involved in the rescues worked for 40 hours straight. Some 97 migrants are missing in that incident.
"Where are Frontex boats in a day like this?" it asked.
"The gun to my head made the difference", a young man called Caleb said after describing how he was forced onto a rubber dinghy late one night and losing sight of his father.