Nohad Machnouk also said he expected more instability linked to the Syrian civil war that has been at the heart of repeated violence in Lebanon over the last four years.
The Nusra Front said on Saturday it was behind the bombing in the Alawite neighbourhood of Jabal Mohsen - an attack Lebanese leaders said aimed to ignite communal strife in a predominantly Sunni Muslim city where long-standing sectarian tensions have been inflamed by the Syrian conflict.
Machnouk said investigators were questioning men who belonged to the same organisation as the two bombers, both of whom have been identified as men from Tripoli.
"The initial information so far says that criminal state of Daesh was the one behind the bombing," Machnouk told journalists in Tripoli, using an pejorative Arabic acronym for the group that has seized wide areas of Syria and Iraq.
Lebanese security officials have warned of plans by Islamic State and the Nusra Front to further destabilise Lebanon. Tripoli, historically a stronghold for Sunni Islamism, is seen as particularly vulnerable.
The last major flare-up in the city, Lebanon's second biggest, was in October, when 11 soldiers and at least 22 militants were killed.
That followed an August attack by militants affiliated to the Nusra Front and Islamic State in Arsal a town on the border with Syria. The militants are still holding some two dozen members of the security forces.
Machnouk said: "Naturally, we expect that all the while the fire in Syria continues, the escalation will increase more and more."
Lebanese flags covered the coffins of eight victims at their funeral and leaders from across the political spectrum have called for unity, warning that the attack aimed to ignite strife.
The families of the two bombers condemned the attack. "Crime has no religion, creed, neighbourhood or family," a spokesman for the families said, adding there would be no mourning for the bombers.
A statement issued on a Twitter feed used by Nusra Front's media arm had said the attack was in revenge for Sunnis in Lebanon and Syria, where the Lebanese Shi'ite group Hezbollah is fighting alongside Syrian government forces.
Hezbollah says it is fighting in Syria to stop jihadists reaching Lebanon. But Lebanese critics say its role there has led to attacks by Sunni Islamist groups in Lebanon.
Lebanon's own sectarian rivalries have been inflamed by the Syrian war that pits the state led by President Bashar al-Assad, an Alawite, against an insurgency dominated by Sunni Islamists.