This makes Alain Guillaume Bunyoni the first person to take on the role in 22 years.
This position was restored by newly elected President Evariste Ndayishimiye.
The Burundian president was sworn into office last week - two months earlier than planned due to the death of his predecessor, Pierre Nkurunziza, earlier this month.
The Members of Parliament (MPs) also approved Prosper Bazombanza as Burundi’s new Vice-President.
This is not the first time the 60-year-old is serving as Vice President. He was Vice President under Mr Nkurunziza between 2014 and 2015.
For the past five years, the new Prime Minister has been the Security Minister of Burundi. He is also an influential figure in the governing CNDD-FDD party. He was a close ally of Mr Nkurunziza during his 15 years in power as president.
The 48-year-old is under US government sanctions for his alleged role in violence and human rights violations during political unrest in 2015.
Burundi exited the International Criminal Court (ICC) in 2017 during an investigation into the unrest that followed a failed coup attempt.