He said his government would “work for the sake of all the people”, adding that the priorities would be reforms in education, health and cutting red tape.
The right-wing nationalist will lead an unprecedented coalition of parties backed by MPs in a vote on Sunday.
He succeeds Benjamin Netanyahu, who was forced out of office after 12 years.
Mr Bennett, leader of Yamina party, will be prime minister until September 2023 as part of a power-sharing deal.
He will then hand power over to Yair Lapid, head of the centrist Yesh Atid party, for a further two years.
Mr Netanyahu – Israel’s longest-serving prime minister – will remain head of the right-wing Likud party and become leader of the opposition.
During Sunday’s debate in the Knesset (parliament) in Jerusalem, a defiant Mr Netanyahu promised: “We’ll be back.”
The new coalition was approved in a razor-thin vote of 60-59, with one abstention.
After the vote, Mr Netanyahu walked over to Mr Bennett and shook his hand.
In his speech, Mr Bennett, 49, said: “This is not a day of mourning. There is a change of government in a democracy. That’s it.