"No tsunami threat exists for other coastal areas although some may experience small sea level changes " it said.
The quake struck at a very shallow depth of only five kilometres (three miles) and was located 340 kilometres (211 miles) east of Kira Kira in the Solomons, the Pacific Tsunami Warning Center in Hawaii said.
The center said a tsunami measuring 0.9 metres (three feet) hit the Solomons following the quake.
The Solomon Islands Broadcasting Corporation cited a witness who said water was covering an airstrip in Lata, in Temotu province, but there were no immediate reports of damage or casualties.
Solomon Islands police in the small town of Kira Kira, on San Chrostobal island, said they felt the quake, but there were no reports of any damage from the quake or a tsunami.
"We felt the shock. We have warned people to get to higher ground," said local police officer Samuel Tora.
The tsunami warning center gave arrival times from a few minutes to several hours to island nations around the South Pacific.
The warning was issued for the Solomon Islands, Vanuatu, Nauru, Papua New Guinea, Tuvalu, New Caledonia, Kosrae, Fiji, Kiribati, and Wallis and Futuna islands.
Residents in the French territory of New Caledonia near the east coast were being advised to evacuate immediately with the tsunami forecast to reach shores around 0300 GMT, according to the French High Commission.
A tsunami watch was also issued for the rest of the South Pacific nations, including Australia, New Zealand and Indonesia. Australia later said there was no tsunami threat to its coastline.
"When no major waves are observed for two hours after the estimated time of arrival or damaging waves have not occurred for at least two hours then local authorities can assume the threat is passed," the Hawaii-based center said.
The Solomons were hit by a devastating tsunami following an 8.1 magnitude quake in 2007. At least 50 people were killed and dozens left missing and more than 13 villages destroyed.