"The accident happened when the bus was descending on a steep hill in rainy conditions," Arusha regional police commander Charles Mkumbo told Reuters by telephone.
"We are still investigating the incident to determine if it was caused by a mechanical defect or human error on the part of the driver."
The students killed in the accident, which occurred at about 9:30am local time in Karatu district, were standard seven pupils, aged 12 to 13, from the Lucky Vincent primary school on their way to visit another school, Mkumbo said.
Tanzania, the second-largest economy in east Africa, has a poor road safety network, but buses remain the main form of public transport between towns.
More than 11,000 people were killed in road accidents in Tanzania between 2014 and 2016, according to government data.