Nkurunziza had announced the bill just after his disputed re-election in July 2015, when his crisis-hit government accused various international groups of using their funding to support the opposition.
The new law, which places tight controls on NGOs' finances, passed yesterday with an overwhelming majority of 105 votes with only one vote against the measure, parliamentary speaker Pascal Nyabenda said.
The law will force international charities and rights groups to keep their accounts in foreign currency at the central bank, with a third of their annual budget to be placed there before the government agrees to cooperate with them.
Burundi's government has been seeking to control cash still coming into its ruined economy as it struggles under the weight of sanctions from major financial backers such as the European Union, its biggest donor, for failing to halt violence in one of the world's poorest countries.
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Controlled by former Hutu rebels, the government has long accused international aid groups of a heavy bias towards members of the Tutsi minority in their recruitment, and the new law states they must "respect the ethnic balance under the constitution".
Nkurunziza's third term run sparked a failed coup attempt and months of protests that led to a government crackdown, armed attacks and assassinations.
More than 500 people have been killed in the current turmoil, and more than 300,000 others have fled the country.
Burundi has also moved to quit the International Criminal Court which was investigating the country, and cut ties with the UN's main human rights body after a damning September report detailed atrocities and warned of "genocide".