Rupinder Singh, a police superintendent, said the driver lost control of the vehicle on a sharp bend and smashed through a protective roadside wall, plunging into the 25-metre-deep gorge.
As many as 200 people were packed into the vehicles.
The travellers were pilgrims visiting the shrine of Ramdevra, a saint revered by both Hindus and Muslims, in neighbouring Jaisalmer district.
Singh told United News of India (UNI) that 80 bodies had been recovered from the crash site and that "casualties were likely to rise as many people were trapped under the trailer."
The Press Trust of India (PTI) said more than 130 people were injured in the crash.
Cranes aided by searchlights were involved in the rescue operation, which UNI said was on a "war footing" and involved ambulances and medical teams from the surrounding area.
Major accidents are common in India where bad roads and overloaded, ill-maintained vehicles are common.
The Delhi-based Institute of Road Traffic Education of India says the country accounts for nearly 10 per cent of all fatal accidents worldwide.