The bus, part of the state-run network in Telangana, was returning from a hilltop temple in the district of Jagtial, about 190 kms (118 miles) from Hyderabad, the state capital, a district official said.
"The driver lost control and the bus skidded off the road and fell into the valley," a top district official, G. Narendhar, told Reuters. "The majority of the passengers have suffered severe injuries."
Media cited brake failure as the possible reason for the tragedy, but Narendhar said that needed to be investigated, adding that the 94 people on board the bus, including its driver and conductor, had exceeded the legal limit of 52.
"The bus accident ... is shocking beyond words," Prime Minister Narendra Modi said on social network Twitter. "Anguished by the loss of lives."
India has the world's deadliest roads, with almost 150,000 people killed in accidents in 2016, the latest government data shows, up 3.2 percent from a year earlier.
