Probir Sikdar, editor of an online news portal, was arrested in Dhaka yesterday after posting that his life was in danger and that the minister should be "held responsible" if he died.
The 55-year-old journalist was taken to the central town of Faridpur, where he was charged under a controversial law and remanded in custody for three days.
"Sikdar was sued under the Information and Communication Technology Act of Bangladesh for defaming a minister on social media," Faridpur police chief Jamil Hasan told AFP.
"This section of the Internet act is draconian and conflicts with Bangladesh's identity as a democratic nation," activist Imran H. Sarker told AFP.
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"It should be abolished as soon as possible to defend the freedom of speech."
A Bangladeshi court last week convicted a publisher of a leading newspaper and his deputy for contempt of court over criticism of a judge.
In a separate case last week the editor of a pro-opposition newspaper was sentenced to three years in jail for failing to declare his assets, in what his supporters said was an attempt to silence a critic of the government.
Home Minister Asaduzzaman Khan Kamal defended Sikdar's arrest, saying the journalist should have gone to police rather than posting on Facebook.
But the Committee to Protect Journalists (CPJ), a New York-based media rights group, demanded Sikdar's immediate release.
"Jailing journalists for publishing threats made against them further undermines confidence in Bangladesh's legal system," said CPJ Asia Program research associate Sumit Galhotra.