Girish Maheshwari was arrested from Bavla near Ahmedabad late on Wednesday night, a senior police official said.
The accused was being brought to Mumbai and would be produced in the Dindoshi court later in the day, he said.
Following a directive from the Union Home Ministry, the police had registered a case against Twitter user '@GirishK1605' for threatening to rape Chaturvedi's 10-year-old daughter.
The case was registered under Section 509 (word, gesture intended to insult the modesty of a woman) of the Indian Penal Code, and relevant sections of the Information Technology Act and the Protection of Children from Sexual Offences Act, on Tuesday after the Congress spokesperson filed a complaint.