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.

Fitzgerald was last season's player of the year as Tigers made the Premiership and Champions Cup semi-finals.
"It's been a great experience playing in the Premiership and in Europe for Leicester," said the 29-year-old.
Fellow forward Bateman, 27, flanker O'Connor, 27, and scrum-half Kitto, 24, have also agreed new deals, the lengths of which have not been disclosed.
Fitzgerald joined Tigers in the summer of 2015 and has made 44 appearances playing at lock and flanker, scoring four tries.
Tigers head coach Aaron Mauger said: "Mike is a very good man to have around the squad. He has a great work ethic in training and on the pitch.
"With the experience he has gained in the last 18 months in English rugby, we believe he will get even better in this environment."
Bateman also arrived that summer, from Exeter Chiefs, and has played 29 times for Leicester.
He achieved the notable feat of playing in all three front-row positions this season.
O'Connor signed in the autumn of 2015, scoring a try on his debut in a European Champions Cup win over Stade Francais, and has made 29 appearances for the club.
Kitto, meanwhile, moved to Welford Road in the summer of 2015 and has played 24 times for Tigers.

The illegal hunting of the animal, and the destruction of where they live, are two of the reasons why.
Conservationists say their population has fallen by more than one third in the past 30 years.
It's been a shock to some of those who care for the creature.
But what makes giraffes so special? Here are five facts you might not have known...
Giraffes have strong tongues that help them to get their food - and to get it fast.
They use their tongues - like humans would use their hands - to grab food. The strength helps them chew food fast to avoid dangerous predators.
Until 2016 giraffes were considered as one species, but now they're considered as four.
They are: the reticulated giraffe, the northern giraffe, the southern giraffe and the Masai giraffe.
They may all look tall and similar, but the creatures are very different - like a polar bear is different from a brown bear.
Some of these have sub-species - where the giraffes are a little different but are still the same type of giraffe.
Just like human fingerprints, no two giraffes have the same spot pattern.
And each of the different species of giraffe have very different types of spot patterns when you look at them closely.
Giraffes spend most of their lives standing up; they even sleep standing up!
They do sometimes give their legs a rest, but all that standing is ok as their legs are very strong.
In fact scientists think that a groove in the giraffe's lower leg bone - which is much deeper than in other animals - helps keep their spindly-legs strong and allow them to support their bodyweight.
The largest giraffe relative ever to have existed is thought to have been the Sivatherium giganteum.
It's believed to have lived one million years ago in Africa and Asia and had a shorter neck and legs than giraffes of today.
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.
[NEXT_CONCEPT]
Leicester Tigers have given new deals to Mike Fitzgerald, Greg Bateman, Brendon O'Connor and Jono Kitto.
[NEXT_CONCEPT]
Giraffes are the tallest mammals on Earth, but their numbers have dropped so low that they're now vulnerable to extinction.