Thousands of people tried to force their way into the temple when its gates opened at daybreak in Deoghar, a town in Jharkhand state, police officer Subodh Kumar said.
Ten pilgrims died and 48 were injured, state Home Commissioner N.N. Pande told reporters. The injured, including four in serious condition, were being treated at a hospital.
He said tens of thousands of people were participating in a monthlong festival at the temple of Shiva, the destroyer, one of the main deities of Hinduism.
"Suddenly there was a push and we all fell down," said Nidhi Kumari, a pilgrim who was waiting to enter the temple. She managed to escape the crush.
Dilip Kumar, another devotee, said he saved himself by running away from the crowds at the temple, 255 kilometers (160 miles) south of Patna, the capital of Bihar state.
Deadly stampedes are fairly common during Indian religious festivals, where large crowds gather in small areas with few safety or crowd control measures.
In July, 27 people died when tens of thousands of pilgrims taking part in a Hindu religious bathing festival triggered a massive stampede on a riverbank in southern India's Andhra Pradesh state.
In October 2013, a stampede in Madhya Pradesh state in central India killed more than 110 people, mostly women and children.