JOIN USSeven children were among the dead, a spokesman for the Ecuadoran Red Cross, Fernando Gandarillas, told AFP.
Rescue officials said around 30 passengers survived, but were injured, some of them badly.
"There's been a very serious accident with approximately 41 people killed," National Transport Council (CNT) director Ricardo Anton told reporters. Earlier reports put the death toll at 35.
Gandarillas said the bus had been travelling from the capital Quito to the eastern towns of Chone and San Isidro, a trip of more than 250 kms, when it fell into a ravine as it negotiated a descent.
A police chief from a nearby town of El Carmen, Diego Iniguez, told media the bus had been filled over capacity and appeared to have experienced a gearbox problem before the driver lost control yesterday.
The national emergency service said the dead included seven children and 13 women. At least 31 other passengers were hurt.
President Rafael Correa regretted the loss of life.
"It's a well-designed highway," he told a Christmas meeting with his staff, adding that the accident was due to "the irresponsibility of the driver who crashed with more than 70 passengers" on board.
"He picked up passengers along the road, which is prohibited," Correa added.
One survivor told television station Ecuavisa that, just before the accident, a woman at the front of the bus started screaming that the driver "can't shift gears."
He said the driver appeared to attempt to keep control of the vehicle, but failed and the bus rolled over for some 200 meters before hitting a tree then dropping into the ravine.
Another survivor, Juan Pablo Alcivar, told Radio Quito how the bus's gears were heard grinding as the driver tried to downshift.
The police traffic accident service said it was at the scene of the carnage to try to determine what went wrong.
The injured were taken to several hospitals in the area, while the bodies were piled up inside police vans and delivered to morgues.