The vote, which followed a motion tabled by the opposition, was defeated by 38 votes to 11 votes, with one abstention.
Earlier, justifying its motion, Vanuatu's opposition listed eight acts which it said had allowed the prime minister's office to be called into question.
It said proposed constitutional amendments were poorly considered and consulted.
It alleged that the government's endorsement of income tax and the PACER Plus trade agreement was also poorly consulted and threatened the livelihoods of ni-Vanuatu and the wider economy.
The opposition also accused the government of interfering in the independent operations of Air Vanuatu, the Shefa provincial government and the telecommunications regulator.
It also said it was questionable for the prime minister to harbour the former deputy prime minister, Joe Natuman, in government for three months after being convicted.
Natuman was ejected from parliament last month by the speaker.
He then resigned having received a two-year suspended sentence in March after pleading guilty to a charge of conspiracy to pervert the course of justice.
In December, Mr Salwai defeated a no confidence motion 37 votes to 13.