Heavily armed members of the group stormed the town of Gamboru-Ngala in Borno, burning a police headquarters, a church and a customs post in the early hours of Monday, residents said.
Shafiu Mohammed, a local resident, told the AFP news agency that the group had overpowered police and customs officers in the town.
"The operation took them two hours. They left around 2:00am (01:00 GMT) without facing any resistance," he said.
Police said a police station was also attacked in Wudil town on the outskirts of Kano.
Baba Mohammed, a Kano police spokesman, said three members of the group were killed in the attack and another 33 arrested.
"An unspecified number of these extremists attacked the police station at around 4:00am (03:00 GMT) and injured two officers, but our men repelled them, killed three and apprehended 33 of them," he said.
Boko Haram was founded in 2004, setting up a base dubbed "Afghanistan" in the village of Kanamma in Yobe, close to the border with Niger.
The local Daily Trust newspaper quoted Ustaz Mohammed Yusuf, who is purportedly the leader of the group, as saying his followers were ready to die to ensure the institution of a strict Islamic society.
"Democracy and [the] current system of education must be changed, otherwise this war that is yet to start would continue for long," he said.
Ogbonnaya Onovo, the national police chief, on Monday vowed to arrest Boko Haram's leaders.
"This is a fanatical organisation that is anti-government, anti-people. We don't know what their aims are yet; we are out to identify and arrest their leaders and also destroy their enclaves, wherever they are," he said.
Felix Onuah, a freelance journalist in the capital, Abuja, told Al Jazeera that the group was largely unknown until now.
"Suddenly they emerged and it was realised that their influence had extended to all of the states in the north," he said.
Sectarian clashes between Muslims and Christians in Bauchi state in February left at least five people dead.
Muslims attacked Christians and set fire to churches in retaliation for the burning of two mosques, which had been blamed on Christians.
Last November, more than 700 people were killed in Jos, the capital of Plateau state, when a political feud over a local election degenerated into bloody confrontation between the two religions.