Hundreds of people had arrived to tackle the fire when a number of containers exploded at the site in Sitakunda.
The cause of the fire is not known, but it is thought that chemicals were stored in some of the containers.
Hospitals in the area are overwhelmed and have appealed for blood donations.
Many of the injured are reported to be in a critical condition with burns covering 60% to 90% of their bodies.
"The explosion just threw me some 10 metres from where I was standing. My hands and legs are burnt," lorry driver Tofael Ahmed told AFP news agency.
Another volunteer told the agency he had seen more bodies inside the area affected by the fire.
Several firefighters are among the killed and injured.
The depot contained millions of dollars of garments waiting to be exported to Western retailers, according to a regional government official. Bangladesh is a major supplier of clothing to the West.