The quake was felt across a wide region after it struck the small, landlocked Himalayan state of Sikkim - which borders Nepal, Bhutan and Tibet - at about 6:10pm (1240 GMT), according to the US Geological Survey.
The epicentre was just over 60 kilometres northwest of the Sikkim state capital Gangtok, where at least 60 people were injured and the town was plunged into darkness by a power cut after the quake.
Tremors were felt in Nepal, Bhutan, Bangladesh and the Indian cities of Guwahati and Kolkata, as well as 1,000 kilometres away in the Indian capital New Delhi.
In Nepal, police said three people were killed, including a motorcyclist and his eight-year-old daughter, when a wall collapsed at the British Embassy compound in the capital Kathmandu, 270 kilometres west of the epicentre.
Telephone landlines to Sikkim, India's least populous state, were knocked out and mobile networks were swamped, making communication with the affected area difficult.
The quake was followed by two strong aftershocks, one with a magnitude of 6.1, and the death toll may rise as reports arrive from remote communities.
Sikkim Chief Secretary Karma Gyatso told AFP that five people had been killed and 60 injured in and around Gangtok as the result of mudslides, building collapses and falling debris.
Nine other people died in India, including one reportedly killed in a stampede by panicked residents in Bihar state and four who were buried when a house fell down near Darjeeling.
In New Delhi, Prime Minister Manmohan Singh called an emergency meeting of the National Disaster Management Authority, and Cabinet Secretary Ajit Seth said that air force planes carrying rescue teams and relief supplies had been despatched to Sikkim.
Strong tremors were felt in Guwahati, the main city in Assam state, some 600 kilometres away, sending people running into the streets.
In Kathmandu, traffic came to a standstill as hotels and bars were evacuated.
In Bhutan, buildings in the capital Thimphu were also rocked.
India's seven northeastern states, joined to the rest of the country by a narrow sliver of land, are located in an area of frequent seismic activity.