The state-run Bhutan Broadcasting Service reported that dozens had been injured.
Lungthen Dorji, the governor of Trashigang district said: "Houses, and monasteries and roads have been damaged. Mobile services are clogged."
According to the US Geological Survey, the epicentre was located just inside Bhutan's border with India, 180km east of Thimpu, the capital.
Strong tremors lasting up to 20 seconds were also felt in Guwahati, the capital of India's northeastern state of Assam. Cracks appeared in several buildings in the city but there was no serious damage, witnesses said.
The quake was also felt in Bangladesh and Lhasa, the Tibetan capital, but there were no reports of damage in either place.
Much of Bhutan, located in the mountains between China and India, is sparsely populated, reachable only by walking paths and without electricity or telephones.
Most of the 600,000 population live by subsistence farming, animal husbandry and forestry.