The bus carrying 51 people overturned just before midnight local time on a motorway connecting the cities of Irkutsk and Chita, the emergency services ministry said in a statement.
"Ten people died and around 20 others were injured," Russia's Investigative Committee said.
The passengers were pilgrims returning from a visit to a convent in the region, the TASS news agency said, citing a local official.
Russia ranks among the world's worst countries for road deaths, mainly due to alcoholism, the poor condition of the roads and failure to follow traffic regulations.