The crash happened on Monday evening on the main road between the capital Naypyidaw and the commercial hub of Yangon.
"It was pouring with rain and the driver lost control and the bus crashed into a bridge and fell onto a rail track," said pol lieut col Min Aung of the Myanmar Police Force headquarters in Naypyidaw.
He said three of the 29 injured were in a serious condition.
The bus was travelling from Naypyidaw to Yangon, a day after a major Southeast Asian summit in the showpiece capital.
There were no foreigners on board, according to Min Aung and the bus operator Yazarmin.
The road between the central city of Mandalay and Yangon -- via Naypyidaw -- has been dubbed the "death highway" by local media due to frequent traffic accidents.
The concrete road was built hastily by the former junta after the generals in 2005 suddenly moved the capital from Yangon to the new purpose-built city erected in scrubland in central Myanmar.
Last month 12 people were killed when a bus collided with a car and burst into flames on the same highway.