More than 50 people, including children and adults, were hospitalised with burns, a doctor at the National Institute of Burn and Plastic Surgery told reporters.
The crash occurred at the Milestone School and College in Dhaka's northern area of Uttara, officials said.
Videos of the aftermath of the crash showed a big fire near a lawn emitting a thick plume of smoke into the sky, as crowds watched from a distance.
Firefighters sprayed water on the mangled remains of the plane, which appeared to have rammed into the side of a building, damaging iron grills and creating a gaping hole in the structure, Reuters TV visuals showed.
"A third-grade student was brought in dead, and three others, aged 12, 14 and 40, were admitted to the hospital," said Bidhan Sarker, head of the burns unit at the Dhaka Medical College and Hospital, where some victims were taken.
Footage also showed people screaming and crying as others tried to comfort them.
"When I was picking (up) my kids and went to the gate, I realised something came from behind...I heard an explosion. When I looked back, I only saw fire and smoke," said Masud Tarik, a teacher at the school.
Muhammad Yunus, head of Bangladesh's interim government, said "necessary measures" would be taken to investigate the cause of the accident and "ensure all kinds of assistance".
"The loss suffered by the Air Force...students, parents, teachers and staff, and others in this accident is irreparable," he said.
The incident comes a little over a month after an Air India plane crashed on top of a medical college hostel in neighbouring India's Ahmedabad city, killing 241 of the 242 people on board and 19 on the ground, marking the world's worst aviation disaster in a decade.
with agencies
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