Burundi has been hit been by deadly protests and a failed coup since President Pierre Nkurunziza announced his bid for a third term in April.
Mr Nkurunziza has been under pressure from both African and Western governments to postpone elections.
Burundi's Foreign Minister Alain Nyamitwe told the BBC that elections would be held by 26 August.
This is when the constitutional limit of the government ends, he said.
One of Burundi's main opposition leaders, Agathon Rwasa, told BBC's Newsday programme that Mr Nkurunziza was a "dictator" who should step down.
Elections could not be held until security improved, a neutral electoral commission was appointed and a crackdown on private media ended, he said.
The international community needed to get involved to help "secure conditions for good elections", Mr Rwasa added.
"We are calling for democracy and not for a dictatorship," he said.
The parliamentary elections had been scheduled to take place on Friday and the presidential poll on 26 June.
BBC World Service Africa editor Richard Hamilton says it looks as though Mr Nkurunziza has finally bowed to outside pressure to postpone these controversial elections - certainly from regional leaders if not Western governments.
However, the move is unlikely to stop the protests which have killed more than 20 people, he says.
Last weekend, regional leaders meeting in the Tanzanian city of Dar es Salaam called for both the delay and an end to the violence.
Opposition parties in Burundi said earlier on Wednesday they would be prepared to resume talks with the government on the crisis.
The opposition says the 51-year-old Mr Nkurunziza's bid to extend his 10-year rules contravenes the constitution, which states a president can only serve two terms.
But Mr Nkurunziza argues that he is entitled to another term because he was first elected by parliament in 2005 - not voters. The Constitutional Court has ruled in favour of the president.
Earlier in May, he survived a coup attempt while he was in Dar es Salaam for talks with regional leaders on the crisis.
The coup was launched by Maj Gen Godefroid Niyombare, a former ally of the president.
His whereabouts are unknown.
Burundi's government has postponed parliamentary and presidential elections that were due this month.