The first operation began early Thursday in the town of Sejnane. Police surrounded the area, arresting 13 people and killing one man who opened fire on them, according to a statement by the Interior Ministry.
Tunisian media identified the man killed as a member of the banned Ansar al-Shariah movement who later pledged allegiance to the Islamic State group.
A second operation was carried out against three homes near the city of Bizerte, 60 kilometers (35 miles) from the capital of Tunis. Two men were arrested while a third fired at police and then took his wife and infant hostage. Police brought in his parents to convince him to give himself up.
Tunisia has been repeatedly targeted by attacks, most recently last month when a young Tunisian gunman opened fire on tourists in the beach resort city of Sousse, killing 37.
Parliament is currently voting on a new anti-terror law that would give police greater powers to combat terrorism, although there are concerns that freedoms won in the country's 2011 revolution are now under threat.