The 16-year-old girl and the 36-year-old married man were in love and killed by the girl's relatives, they said.
The bodies were later cremated. One man has been arrested.
So-called honour crimes are common in India, where many still prefer arranged marriages within their own caste and love matches are often frowned upon.
In 2011, India's Supreme Court said those involved in honour killings should face the death penalty.
The latest incident took place in Amaitha village in Bihar's Gaya district on Wednesday.
Reports said the couple had eloped a few days ago, but were found by the girl's family and brought back to village on Wednesday morning.
"A married man, Jairam Manjhi, had fallen in love with a girl. The girl's family killed the couple by brutally beating them and then burnt their bodies," senior police official Manu Maharaj told BBC Hindi.
Mr Maharaj said the village council held a meeting to resolve the issue, but the girl's family refused to see reason and killed the couple.
Police have registered a murder case against six people, but so far only one man had been arrested.
A couple who had eloped in the northern Indian state of Bihar have been lynched by villagers in what is suspected to be a case of honour killing, police said.