City officials say that the factory was used to produce and store high-explosive bombs for the military.
They add that the processing facility at the JSC Kristall Research Institute plant has been completely destroyed by the blast.
A factory official says five people were inside at the time, but they were safely evacuated.
Most of the people who were hurt were cut by flying glass from the explosion, which also caused a shockwave that smashed windows in homes and other factories in the city.
Dzerzhinsk city officials have declared an emergency in the surrounding areas, while the Investigative Committee of Russia says it has launched a criminal investigation into potential safety violations at the plant.
A local health ministry statement says: "According to the latest information, 79 people asked for medical help after the explosion at Kristall: 38 factory workers and 41 residents of the city. There are no children among the injured."