The oil spilled after a tanker carrying 350,000 litres (77,000 gallons) collided with another vessel, spreading a sheen across 60km of waterways.
Footage shows villagers scooping oil from rivers and birds covered in black liquid in the Sundarbans area.
The Sunderbans is a Unesco heritage site, home to many rare species.
The oil leaked into the river at a sanctuary for rare dolphins.
Environmentalists have expressed concern that the impact on wildlife could be disastrous.
But government officials say they still cannot give an assessment of the likely damage.
The UN on Thursday called for a complete ban on all commercial vessels using the waterways of the Sunderbans.
The Bangladeshi navy initially sent four ships to deal with the spill, and planned to use chemicals to disperse the oil.
But it was unclear whether the chemicals could be used, amid fears they could cause further damage.
"This catastrophe is unprecedented in the Sundarbans and we don't know how to tackle this," Amir Hosain, chief forest official of the Sundarbans, told AFP news agency.
"We're worried about its long-term impact, because it happened in a fragile and sensitive mangrove ecosystem."The tanker was reportedly on its way to deliver the oil to a power plant in Gopalganj when it was struck by another vessel.
The Sunderbans, a Unesco world heritage site, is a vast river delta on the northern shore of the Bay of Bengal.
Its mangroves and rivers are home to a vast array of plant and animal life, much of it unique to the region.
The government declared areas in the southern Sunderbans to be a dolphin sanctuary in 2011, after research suggested some 6,000 Irrawaddy dolphins lived in the area.
