Police spokesman Pierre Nkurikiye said one policeman was among the dead and another was seriously wounded in clashes with "insurgents" in the Ngagara district of Bujumbura.
The clashes are the latest in months of unrest triggered by Nkurunziza's successful bid to win a third term in office.
Opponents said the bid breached the law and the terms of a peace deal that ended a 13-year civil war in 2006.
Residents said the clashes began when soldiers stormed the neighbourhood.
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The UN warned last month that Burundi risked sliding back into civil war after a dramatic rise in killings, arrests and detentions, with almost 200,000 refugees fleeing to neighbouring nations.
"The unstable situation in Burundi is marked by human rights violations including extra-judicial executions, arbitrary arrest and detention of persons opposing the third term," a UN report this month read.
"Due to ongoing violence, threats and intimidation towards opposition supporters and fear amongst communities of further violence, displacement is likely to continue."
Street protests and violence have ebbed and flowed in Bujumbura since April when Nkurunziza launched his third-term bid culminating in his controversial re-election in July.