Gen Niyombareh, in a statement read to reporters in a military base, said he did not recognise the leadership because the president's bid for a third term violated the constitution.
Soldiers surrounded the offices of the national broadcaster in the capital, Bujumbura.
In a radio broadcast, Gen Niyombareh said: "The masses have decided to take into their own hands the destiny of the nation to remedy this unconstitutional environment into which Burundi has been plunged. "The masses vigorously and tenaciously reject President Nkurunziza's third-term mandate... President Pierre Nkurunziza has been relieved of his duties. The government is overthrown."
Gen Niyombareh's "national salvation committee" comprises at least five other army and police generals.
He said in the radio broadcast that the mission of the body was the "restoration of national unity... and the resumption of the electoral process in a peaceful and fair environment". One witness told Reuters that crowds who had earlier been out on the streets of the capital protesting against the president were now cheering and celebrating his apparent dismissal.
The unrest began on 26 April and has led to the deaths of more than 20 people. President Nkurunziza has rejected calls to postpone next month's election.
Tens of thousands of Burundians have fled to neighbouring states in recent weeks.