About 50 other people were injured and the fire in Dhaka was mostly under control after more than nine hours of frantic efforts by firefighters.
The Chawkbazar area is crammed with buildings separated by narrow alleys.
The neighbourhood is a mix of residential and commercial, with buildings that commonly have shops, restaurants or warehouses on the ground floors.
The blaze started late on Wednesday night in one building but quickly spread to others, fire department director general Brigadier General Ali Ahmed said.
The death toll rose to 70, with many of the victims trapped inside the buildings, said Mahfuz Riben, a control room official of the Fire Service and Civil Defence in Dhaka.
He said: ‘Our teams are working there but many of the recovered bodies are beyond recognition.
Our people are using body bags to send them to the hospital morgue, this is a very difficult situation.’ Some floors of the destroyed buildings had chemicals and plastic in storage.
Most buildings are used both for residential and commercial purposes despite warnings of the potential for high fatalities from fires after one had killed at least 123 people in 2010.
Authorities had promised to bring the buildings under regulations and remove chemical warehouses from the residential buildings.
The death toll could still rise as the condition of some of the injured people was critical, said Samanta Lal Sen, head of a burn unit of the Dhaka Medical College Hospital.
Mr Sen said at least nine of the critically injured people were being treated in his unit. Witnesses told local TV stations that many gas cylinders stored in the buildings continued to explode one after another.
They said the fire also set off explosions in fuel tanks of some of the vehicles that got stuck in traffic in front of the destroyed buildings.