A police spokesman said the incident took place in Veesu in Qazigund when the bus, registered in Uttar Pradesh, was going to Jammu from Srinagar.
The deceased was identified as Ganga, a man from Andhra Pradesh, he said.
The spokesman said nine of the injured were being treated at the Emergency Hospital, Qazigund, while the rest were shifted to the District Hospital, Anantnag.
The pilgrimage to the Himalayan cave shrine began on June 29.