The 40-seater bus carrying Hindu pilgrims from Sawai Madhopur to Lalsot veered off a bridge and plunged into river Banas.
The driver apparently lost control while attempting to pass another vehicle in foggy conditions.
The accident occurred at Dubi in the state's Sawai Madhopur district this morning. Most of the people died due to drowning including the bus driver.
Death toll may rise, police said. The identities of the deceased have not been ascertained as of yet and the injured have been rushed to a nearby hospital.
The wreckage of the bus was pulled out of the river with a crane and gas cutters. Soorwal police station SHO Anoop Singh said that of the 33 people killed, seven are women and four children.More news: Armed Forces Bowl: 3 takeaways from San Diego State-Army
Out of the seven injured, one is said to be critical.
Prime Minister Narendra Modi said in a statement that he was "anguished" by news of the crash.
He also appealed to the state government to provide all possible help to the injured.
Rajasthan Chief Minister Vasundhara Raje took to Twitter to express her condolences and support for the victims.