Mr Sahoo, the MLA from Basta in Balasore district, received the call on his mobile phone from an unknown number around 8.30 am on Tuesday.
The cyber cell of the Balasore police started an investigation and Sk Mujamil was arrested from the Mangalpur area, an officer said.
He has been booked under various sections of the Indian Penal Code.
During the investigation, it has been found that the suspect is a habitual offender and he had made a similar nuisance earlier with another MLA.
Sk Mujamil used to download the phone number and images of the legislators through the internet randomly and make such threat calls to extract money, Superintendent of Police Sudhansu Sekhar Mishra said.
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