Local residents said the person killed was an official with the National Liberation Forces (FNL), a former rebel group that had been the ruling party's main competition, until it pulled out of the election over fraud claims.
Another man was shot dead in Bujumbaura's Musanga neighbourhood in an attack suspected of being politically motivated, a local official said.
At least eight people have been killed since opposition candidates pulled out of the race. The FNL has come under suspicion for organising the attacks, with the police searching the home of the party's leader, Agathon Rwasa, on Friday night.
Speaking on condition of anonymity, a police official confirmed the search was linked to the violence.
"We have just carried out a search at Agathon Rwasa's home as part of our investigation into the spate of grenade attacks rocking the country," the official said.
"Several suspects who were arrested with grenades have cited Rwasa as the planner of this unrest ... so we are looking for clues," he said.
The FNL has denied involvement in the attacks, and said that their leader was being targeted for political reasons.
Overnight, police also arrested six members of the opposition group the Movement for Solidarity and Development (MSD) in the capital.
"Last night at around 10pm [20:00 GMT], dozens of policemen arrested six of our party's top officials ... as they were enjoying a drink in a bar," Alexis Sinduhije, chairman of the MSD, told AFP news agency.
A police spokesman said the men were named by an informant, who said they intended to disrupt the presidential poll.