Brig Gen Chris Olukolade told the BBC the aim was to assert Nigeria’s “territorial integrity as a nation”.
Earlier, a 24-hour curfew was imposed in parts of the city of Maiduguri, where the military said it had arrested 65 “terrorists”.
Maiduguri has been an important base for Boko Haram Islamist militants.
Earlier this week, Nigerian President Goodluck Jonathan declared a state of emergency in three north-eastern states – Borno, Adamawa, and Yobe – after a series of deadly attacks by militant groups.
Nigerian military spokesman Brig Gen Olukolade said the army would continue its operation “as long as it takes to achieve our objective of getting rid of insurgents from every part of Nigeria”.
He told the BBC World Service Newshour programme that the offensive “knows the targets it is after, and it took a long time to plan and prepare for this”.
“It is their (insurgent) bases, their weaponry, their logistics that we are going to deal with in this operation,” he added.
Earlier, an army statement named 12 areas of Maiduguri, the capital of Borno state, which would now be under permanent curfew.
Correspondents say the neighbourhoods listed are considered strongholds of Boko Haram. The whole of Maiduguri was already subject to an overnight curfew.
Witnesses said troops were stopping lorries from entering the city.
The army also said the 65 insurgents had been arrested trying to infiltrate Maiduguri. Those held had been “fleeing from various camps now under attack”.
However, there has been no independent confirmation of the arrests.
The army said 10 suspected insurgents were killed in clashes with troops in Maiduguri on Friday and weapons were seized including rocket-propelled grenade launchers.
Nigerian war planes and helicopter gunships also attacked several militant training camps in the north-east on Friday, officials said.
One plane was hit by anti-aircraft fire but officials said it had returned to base safely while the “terrorist base” was “completely destroyed”.
US Secretary of State John Kerry has urged the Nigerian army to show restraint and not violate human rights as it pursues the militants.
Mr Kerry said there were “credible allegations” of “gross human rights violations” by the Nigerian military.
Last November, Amnesty International accused Nigeria’s security forces of carrying out widespread abuses in their campaign against Boko Haram, including extra-judicial killings, enforced disappearances and torture.
The BBC has also received reports of civilians fleeing remote areas in Borno State and arriving in towns on the border with Cameroon.
Cameroon shares hundreds of miles of border with Nigeria and the area adjacent to Borno is considered porous.
Although Cameroon’s security forces have stepped up security in the region, a source told French radio that the government in Yaounde had been given no advance warning of the Nigerian campaign.
More than 2,000 people have died in violence in Nigeria since 2010, most of which is blamed on Boko Haram.
The group, whose name means “Western education is forbidden”, says its quest is to overthrow the Nigerian government and create an Islamic state.
There has been growing concern that Boko Haram is receiving backing from al-Qaeda-linked militants in other countries.