Jugnauth, 86, officially handed his resignation to President Ameenah Gurib Fakim - whose role is ceremonial - after long hinting he would step down before his term expires in 2019.
"The job of prime minister involves great responsibility. It is a great burden. I have carried it but now it is time to make way for the youth," he said after resigning.
His son Pravind, 55, was then handed a letter nominating him to the post ahead of a swearing-in ceremony later today.
The younger Jugnauth is both finance minister and leader of the ruling Militant Socialist Movement (MSM). Constitutionally, it is he who takes over if the prime minister resigns.
However the opposition has rejected his "inheritance" of the position, and will be boycotting his inauguration.
"The best would have been for the prime minister to dissolve the national assembly before his resignation and call an early election," said opposition leader and former prime minister Paul Berenger.
Jugnauth has done several stints as premier since 1982 and one in the ceremonial role of president.
"It is a shame he has ended more than 50 years of political career in this way. We will organise a political campaign across the country to demand the holding of general elections," Berenger said.
Mauritius, an island of 1.3 million inhabitants, regularly tops the Mo Ibrahim Index of African Governance, and has held a series of peaceful elections and smooth handovers of power since independence from Britain in 1968.
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