Shawkat Afsar of Islami Chhatra Shibir, student wing of the Jamaat, was arrested on a tip-off, Deputy Commissioner of the Dhaka Metropolitan Police Masudur Rahman said.
Afsar was arrested from a coaching centre, Rahman was quoted as saying by The Daily Star.
Yesterday, the embassy received a letter from an unidentified sender asking the Egyptian authorities to stop "killing of Muslim Brotherhood members" in Egypt.
The authorities later decided to shut down the consular section of the Egyptian embassy for 11 days from today.
The law enforcement agencies have since beefed up security measures on and around the embassy.
Several right-wing Islamist groups earlier protested the Egyptian army crackdown on the Brotherhood activists.
Some 200 student wing activists of the Jamaat, believed to have close political affiliation with Brotherhood, staged a protest march in Dhaka yesterday.
They exploded several crude bombs to denounce the army actions in Egypt.
Dhaka and Cairo maintain a close diplomatic tie but Dhaka is yet to issue any statement on the recent developments in Egypt.
