The centre issued warnings for Solomon Islands, Vanuatu, Papua New Guinea, Nauru, New Caledonia, Tuvalu and Kosrae.
It said people near impacted coastal areas should stay alert and exercise normal caution near the sea.
American Samoa had issued a tsunami watch following the earthquake but that was later cancelled.
The quake, which hit just before 5am local time on Friday morning, was initially reported to be of magnitude 8 by the US Geological Service but was downgraded shortly afterwards.
The National Disaster Management Office Solomon Islands said it has been trying to contact the affected areas.
Its Director, Loti Yates, said his office is unsure about the impact of the quake at this stage as it’s still early in the country but they should have more information in the coming hours.
The epicentre of the quake was off Kirakira, capital of the Makira and Ulawa province and caused people in the capital Honiara and outlying regions to run for the hills
There are no reports of major damage but it did cause a power blackout to some areas.
Police commander at Makira-Ulawa province John Smith said the quake caused real fear in his neighbourhood.
'It was quite a big earthquake and people were panicked because it happened when it was still quite dark and most of the people run to the hills."
RNZ International's correspondent in Solomon Islands, Dorothy Wickham, said many people in Honiara are now starting to head home and those with businesses are checking their properties for damage.
"There's been no reported damage here that I'm aware of yet but I will take a ride around town a little later," she said.
"I spoke to the police in Kirakira about an hour after the quake and they said so far, no damage to permanent buildings but there have been reports of leaf thatched houses collapsing under the quake and a lot of people have fled into the hills because the epicentre was closer to Kirakira."
Ms Wickham said radio services were still playing automated music at that time of the morning and there needs to be a better system for civil defence officials to communicate with residents as she only knew about the tsunami risk from the internet.
GNS science duty seismologist Geoff Kilgour said the quake was about 40km below the surface.
It appeared to be a reverse fault, where the land was pushed up, which is why tsunami waves may be generated.
"It's very early at this stage to talk about the magnitude of the tsunami," he told RNZ's Morning Report
An 8.0 quake off Solomon Islands this morning. Photo: Pacific Tsunami Warning Center