Salwai, from the Reunification of Movements for Change party, was announced as leader some three weeks after more than 200,000 voters went to the polls, the Vanuatu Daily Post and other media reported.
It followed parliament being dissolved in November by President Baldwin Lonsdale when 14 lawmakers were jailed for bribery.
The political breakdown came after a period of instability with four changes of prime ministers in the past four years.
Salwai, a francophone, thanked local chiefs, church leaders and the people of Vanuatu for their role in "steering the country back on track after a very tough year in 2015".
More From This Section
"The changes which you have shown in your voting have shown that you have voted for change.
"Today we have two thirds of the (52-member) parliament who are new members of parliament... And this has shown that this house must recognise the voice of the people."
The constitutional crisis erupted last year when the 14 lawmakers were convicted on bribery charges and hastily tried to pardon themselves.
Parliamentary speaker Marcellino Pipite, using his powers as acting president while Lonsdale was overseas, issued pardons for himself and his co-conspirators.
The charges centred on payments of 35 million vatu (USD 312,000) made by Deputy Prime Minister Moana Carcasses to 13 other politicians while they were all in opposition.
Vanuatu, which gained independence in 1980, is still recovering from a deadly category five storm last March that destroyed homes and crops and contaminated water supplies.