Blasts and gunfire were heard across the city during the simultaneous attacks on police posts and vehicles.
The attacks took place in districts that have seen protests against President Pierre Nkurunziza's intention to seek re-election.
Since his announcement in April Burundi has also seen a failed military coup.
Police have blamed the attacks on opposition activists. A senior police official told AFP one of the officers was seriously injured.
It is possible that violence will escalate ahead of parliamentary and presidential elections next month, says the BBC's Africa Editor Mary Harper.
Burundi's main opposition leader has said he will participate in next month's presidential election if certain conditions are met.
Rights groups say at least 70 people have been killed and 500 wounded since the protests began in April, with more than 100,000 fleeing the country.
The opposition and civil societies say Mr Nkurunziza's attempt to run for office again contravenes the constitution, which states a president can only serve two terms.
However, the Constitutional Court has ruled that he was first elected in 2005 by parliament, not in a popular vote, and so he is eligible to stand again.
Security officials in Burundi say 11 policemen have been injured in overnight grenade attacks in the capital Bujumbura.