The bodies of three people who drowned in the accident were found on Friday, a spokesman for the Tunisian defence ministry said, adding the boat left Zuwara on the northwestern Libyan coast on Thursday.
A fishing boat picked up 16 survivors, who were transferred on board one of three military vessels involved in the search and rescue operation, said the spokesman Mohamed Zekri.
A Maltese helicopter was also mobilised, he said.
According to those rescued between 60 to 70 people from sub-Saharan Africa were thought to be on board, Zekri added.
Interior ministry spokesman Sofiene Zaag said however around 75 people were believed to have been on board the boat bound for Italy when it departed Libya.
Italy's far-right Interior Minister Matteo Salvini has vowed to close the country's ports to migrants, although earlier Friday dozens who had been rescued at sea disembarked in Sicily.
Rome's populist government has taken an increasingly hard line on migration, and Salvini, head of the anti-immigrant League party, last month signed a new directive banning charity vessels from rescuing migrants off Libya.
The Mediterranean is the "world's deadliest sea crossing" according to the United Nations refugee agency, with one person in 14 dying on the route from Libya to Europe last year.
The Tunisian Forum for Economic and Social Rights NGO denounced in a statement what it said were the European Union's "restrictive and inhumane policies" which it said led to the latest "human tragedy".
Libya, which has been wracked by chaos since the 2011 uprising that killed veteran dictator Moamer Kadhafi, has long been a major transit route for people desperate to reach Europe.
Over the past week, and despite a surge in violence following the launch of an assault on Tripoli by strongman Khalifa Haftar, Libyan authorities have rescued hundreds of migrants stranded at sea.
