Assailants threw grenades in both the capital Bujumbura and at some provincial voting centres ahead of Monday's parliamentary and local elections, delaying the start of voting in many areas, police and election officials said.
Another grenade exploded in the capital shortly after voting began.
UN Secretary General Ban Ki-moon has called for the elections to be delayed after the opposition said they would not take part, as Burundi faces its worst crisis since its civil war ended nine years ago.
Police patrolled the streets of Bujumbura, especially in opposition areas where the worst violence was seen during weeks of protests. In those areas, few civilians were seen at polling stations.
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Stations for the parliamentary and local elections opened late in some areas, although election commission spokesman Prosper Ntahorwamiye insisted that, apart from some delays due to the violence, voting was "going well".
"Voting has not yet begun in many centres in the capital because election officials are trying to prepare materials and in almost all of the stations, these arrived late because of the overnight attacks," said election commission chief in Bujumbura, Cyriaque Bucumi.
Burundi was plunged into turmoil in late April when Nkurunziza launched his drive for a third consecutive five-year term, triggering widespread protests.
Opponents say his bid for another term is unconstitutional and violates a peace accord that paved the way to end 13 years of civil war in 2006. Presidential polls are due on July 15.