Nguyen Hoai Nam, 44, admitted to the police that he posted the false information and threatened local authorities to "attract attention on social media."
He faces charges of "illegally posting or using information on computer networks and telecommunication networks," an offense that carries a maximum prison term of seven years.
Investigators said he made the claim several times on his Facebook page between April 13 and 18 that Vietnam had seen its first deaths due to Covid-19.
"Vietnam has recorded three Covid-19 deaths and the government is concealing this," he wrote.
He also insulted local police and other authorities and threatened to stab them if they came to his house.
Since the pandemic broke out in late January, authorities have slapped fines of up to 15 million ($427- 641) on hundreds of individuals for posting fake news.
To tighten control over the propagation of fake news, authorities are slapping criminal charges on those posting false information about the pandemic with attendant jail terms of up to seven years.
A man in the Central Highlands province of Lam Dong is being investigated for posting false information on his Facebook page that Da Lat has recorded three infections, with one patient dying.
Vietnam has recorded no new cases for the last four days.
Of the 268 patients diagnosed so far, only 64 are still in hospital after the remaining 204 were discharged. No dealths have been recorded to date.
The pandemic has spread to 210 countries and territories, with more than 165,000 deaths recorded so far.