Burundian officials and witnesses said the armed group crossed into Cibitoke province during the night, and that Burundian soldiers and police had been locked in battles against the group throughout the day.
"Eight attackers have already been killed," the governor of Cibitoke province, Anselme Nsabimana, told AFP.
A local witness said residents had seen around 200 fighters armed with assault rifles and grenade launchers cross into the country.
Previous attacks in the border area have been claimed by a splinter faction of the National Liberation Forces (FNL), whose full name is Party for the Liberation of the Hutu People.
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But the rebels who still fight on have claimed a strong of attacks this year, and vowed to "intensify" their attacks ahead of presidential elections in June 2015.
Burundi, a small nation in Africa's Great Lakes region, emerged in 2006 from a brutal 13-year civil war and its political climate remains fractious ahead of the polls.