Director General of Drug Administration Maj Gen Mahbubur Rahman announced the approval on Thursday.
The first shipment of 500,000 doses of the Sinopharm vaccine will arrive in Bangladesh within two weeks, he added.
The World Health Organization expects to decide whether to give emergency approval for China's two main COVID-19 vaccines in the next two weeks, Assistant Director-General Mariângela Batista Galvão Simão told a briefing on Monday.
Simao said the WHO could decide on a vaccine made by Sinopharm by the end of this week, and one made by Sinovac Biotech by the end of next week.
China has already deployed millions of doses of both vaccines at home and has exported them to many countries, particularly in Latin America, Asia and Africa.
An emergency listing from the WHO is an indication to national regulators of a shot's safety and efficacy, and would allow the Chinese vaccines to be included in COVAX, the global programme to provide vaccines mainly for poor countries.