The patients, who were using ventilators in the hospital's intensive-care unit, died after the fire triggered an evacuation, said Nazmul Haque, director of the hospital, Reuters reported.
It was not immediately clear what caused the blaze, a fire service official said.
Hospitals are struggling to deal with a fresh spike in coronavirus infections this month in Bangladesh, which has reported 559,168 cases and 8,571 deaths.
Lax regulations and poor enforcement have often been blamed for large fires in the South Asian nation that have killed hundreds of people in recent years.