The epicentre was 29 miles to the east of Taron on Saturday, the US Geological Survey (USGS) said.
A strong aftershock hit the area of the original quake within the space of an hour, with the USGS reporting 5.5 and 6.3 magnitude quakes.
There are no reports of casualties.
The USGS added: “Overall, the population in this region resides in structures that are vulnerable to earthquake shaking…some casualties and damage are possible.”
People living in Bougainville fled to higher ground. Local resident Christabel Biasu said: “The town residents have vacated the whole place; those in the villages live higher up, so they’re OK, it’s just those near the coast.”
A nurse at Buka General Hospital in Bougainville said the quake was so strong it felt like the building she was sleeping in would topple.
She said patients were being moved a few kilometres to higher ground.
The Pacific Tsunami Warning Centre had warned of hazardous tsunami waves but it later said the threat was over.
Tsunami warnings for Indonesia, the Solomon Islands, Pohnpei, Chuuk, Nauru, Kosrae and Vanuatu were also cancelled.
Papua New Guinea sits on the Ring of Fire, the arc of seismic faults around the Pacific Ocean where earthquakes are common.
The neighbouring Solomon Islands were hit by a devastating tsunami in 2013 after an 8.0-magnitude earthquake struck the region.
At least 10 people died and hundreds of homes were destroyed leaving thousands of people homeless.