Police said the pilgrims were going to Haridwar after offering prayers at the Vaishno Devi shrine in Jammu and Kashmir.
They all belonged to Bhagalpur district of Bihar.
Bus driver Gaund Das and conductor Janardhan Yadav suffered fractures in the accident, according to Senior Medical Officer (incharge) Dawinder Singh of local Civil hospital.
Both hailed from Behrampur. Mr Yadav was referred to a Jalandhar hospital as his condition was critical.
One of the injured pilgrims Anil Kumar told newsmen that they were all asleep when the accident occurred. "We don't know how the accident occurred", he said.
He said there were about 20 women and four children in the group of pilgrims.
SHO Sukhpal Singh said two persons of the second bus, including its conductor, also sustained minor injuries. However, they neither reported to police nor at the hospital and drove away after the accident, he said.
Phagwara SDM Jyoti Bala Mattu visited the hospital to oversee treatment of the injured pilgrims.