One of presidential adviser Zenon Ndaruvukanye's bodyguards was killed and another seriously injured in the attacks, said deputy police spokesman, Moise Nkurunziza, to AP by phone.
Bujumbura has been wracked by violence since April when President Pierre Nkurunziza defied street protests to seek and eventually get a third term that many saw as unconstitutional.
Both opponents and supporters of the government have been killed in apparent revenge attacks.
Ndaruvukanye, a member of the ruling party, was on his way from home to his office when six gunmen fired at his vehicle in the Kajaga neighborhood. After spraying the vehicle with bullets, the gunmen fled on a pick-up truck.
Local human rights activists say the government is facing an armed rebellion, possibly by renegade soldiers, but officials say those behind the attacks are just criminals. More than 240 people have been killed in sporadic violence, according to the UN.
"Dying in Burundi has become very easy. Dead bodies litter streets daily and the government is not bothered," he said. Burundi has a history of deadly conflicts between the country's Hutu and Tutsi ethnic groups.
Nkurunziza took power in 2005 near the end of a civil war in which some 300,000 people were killed between 1993 and 2006.
