The quake hit at 04:35 local time (23:05 GMT Sunday) about 29km (18 miles) northwest of Imphal, the capital of Manipur state, according to the US Geological Survey (USGS).
Strong tremors have been felt across the region.
The earthquake was originally reported to have measured 6.8 magnitude.
The tremor cracked walls and a newly built six-story building in Imphal collapsed, police said. Other buildings were also reported to have been damaged.
At least five people have been killed in Manipur and more than 30 injured, the Press Trust of India news agency reported.
India's Meteorological Department said the quake struck at a depth of 17 kilometres (about 10 miles).
Panicked residents also fled into the streets in Bangladesh, said the BBC's Salman Saeed in Dhaka.
Deepak Shijagurumayum, a resident of Imphal, told the AFP news agency by phone that his house was severely damaged by the quake.
"Almost everyone was asleep when it struck and were thrown out of their beds," Mr Shijagurumayum said.
"People were crying and praying in the streets and in open spaces. Hundreds remained outdoors for several hours fearing aftershocks."
An official at a hospital in Imphal said more than 50 people had been admitted since the quake with head injuries and fractures.
Shaking was felt as far away as Kolkata, 600 kilometres (370 miles) away. "Many people were seen coming out of their homes in panic," local resident Rabin Dev told AFP news agency.
Indian Prime Minister Narendra Modi tweeted that he had spoken to the region's chief ministers and federal home minister Rajnath Singh "on the situation arising in the wake of the earthquake".
Casualties have not yet been reported on the Myanmar side of the border, which is sparsely populated.
The region has a history of powerful earthquakes caused by the northward collision of the Indian and Eurasian plates. They are moving towards each other at a rate of 4-5cm per year.
In 2005, a magnitude 7.6 quake in Pakistan-administered Kashmir left more than 75,000 people dead.
In April 2015, Nepal suffered its worst earthquake on record with 9,000 people killed and about 900,000 homes damaged or destroyed.
An earthquake measuring 6.7 magnitude has hit northeast India, near its borders with Myanmar and Bangladesh, killing at least five people.