The pre-dawn accident happened in the Khed area of Ratnagiri district in Maharashtra state, 350 kilometres (220 miles) south of the state capital Mumbai.
"The bus was speeding and the driver probably lost control at the turning ahead of the bridge. It is unclear whether the driver was drunk," Mahendra Singh Pardesi, head of the Khed police station, told AFP by telephone.
The bus was travelling from the holiday state of Goa to Mumbai.
Most of the injured, including the driver, sustained head and arm injuries and were undergoing treatment at a local hospital, he said.
Pardesi said there was one foreign tourist on the bus, a Russian woman who escaped without serious injuries.
Images from the scene showed the bus lying on its roof next to a bridge. The vehicle had landed on a dried-up patch of land next to low-lying water in the Jagbudi River.
About 110,000 people were killed in Indian road accidents in 2011 -- more than 300 every day -- according to the last available statistics from the National Crime Records Bureau.
Bad roads, speeding vehicles and poor driving were among the contributing factors, and bus crashes with a double-digit death toll are far from rare.
Tuesday's crash came a day after a senior official revealed that more than 800 people have died in accidents on the Mumbai-Goa highway in the last three years.
Maharashtra's home minister R. R. Patil, in a written reply to a question, told the state legislature Monday that 828 people died and 2,411 were seriously injured between 2010 and the end of 2012, the Press Trust of India reported.
The World Health Organisation's global status report on road safety 2013 found that eight percent of India's road user deaths were drivers or passengers of buses, while 32 percent were riders of motorbikes or three-wheelers.