Speaking after filing his official candidacy, he denied suggestions that violence was spiralling out of control.
He said there was no need to halt elections, despite violent protests sparked by his presidential bid.
Eighteen people have died in the recent unrest, the worst in Burundi since a civil war ended in 2005.
"Burundians have no problem with elections... because 99% of the country is peaceful," Mr Nkurunziza said.
"These demonstrations have turned into insurrection, but it is something that will be controlled... and I assure you that the elections will go well."
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The president compared his country's situation with Nigeria, where he said polls had gone ahead in April despite there being "no security" because of the insurgency by Boko Haram Islamist militants.
The head of the electoral commission Pierre Ndayicariye echoed the president's position, telling the BBC that the polls, due to be held in June, will not be postponed.
The UN said on Friday that more than 50,000 Burundians have fled their country since April because of fears over pre-election violence.
Mr Ndayicariye urged those who have fled "to come back, because their vote is very important in ensuring a democratic Burundi".
Burundi's constitutional court ruled on Monday that Mr Nkurunziza can run for a third term.
AU commission chief Nkosazana Dlamini-Zuma said on Thursday that elections could not be held in Burundi in the current climate, and expressed doubts about whether the move was constitutional.
The US has accused Mr Nkurunziza of violating the peace accord which ended the brutal 12-year civil war by seeking re-election.
Speaking after a closed-door meeting on Burundi at the UN Security Council on Friday, America's UN ambassador Samantha Power said the US was considering "targeted measures including visa bans or sanctions" against those involved in organising or taking part in violence against protesters.
Opposition and civil society groups insist that a third-term bid is unconstitutional, but the president argues his first term does not count as he was appointed by parliament, not directly elected by the people.
Mr Nkurunziza, a former rebel leader, has ruled Burundi for nearly 10 years.
Burundi's leader Pierre Nkurunziza has launched his third-term bid in the capital Bujumbura, defying criticism from the African Union (AU) and the US.