The man, identified as 39-year-old Abdur Rahim Biplobi, was apprehended outside a mosque in Dhaka's South Keraniganj on Thursday.
The incident in Cumilla touched off a series of attacks in Chandpur on Oct 13, leading to several fatalities, the CID said in a statement.
"In such a tense situation, Islamic speaker Abdur Rahim Biplobi made a provocative speech in a sermon the same day."
Biplobi blamed the police for the incidents in Cumilla and Chandpur while making "insulting, insolent and contemptuous remarks", according to the CID.
A case has been filed against him under the Digital Security Act with Paltan Model Police.