The US Geological Survey downgraded the quake, which struck at 6:38am New Zealand time to magnitude 7.8 from an original reading of 8.0. It put the depth at about 40km.
The US-based Pacific Tsunami Warning Centre (PTWC) had issued warnings for the Solomon Islands and neighbouring island chains of a potentially hazardous tsunami, but a couple of hours after the quake it said the threat had passed.
An initial tsunami watch alert for Hawaii had earlier been cancelled.
New Zealand Civil Defence issued a nationwide marine and beach alert, but that was cancelled around 8.30am.
Even though the tsunami threat to New Zealand has been cancelled Civil Defence is still warning people to keep out of the water.
Strong and unusual currents as a result of the tsunami are expected for the rest of the day, especially along the entire west coast.
John Pirimare, a resident on Nafinua Island, spoke to Reuters from up the hills where he evacuated immediately after the quake with around 500 to 600 villagers after they received the tsunami warning on their phones.
Yates said the authorities in the Solomons would start to assess the damage as morning dawned.
Some 50,000 people live in Makira Province, the closest to the epicentre of the quake.
"It's not yet clear what sort of impact this has had," Yates said.
"It was dark so it is just now getting light and we can try and assess the damage."
James Samani, duty manager at the Solomon Kitano Mendana Hotel in the capital Honiara, said the earthquake was strongly felt but the hotel was not damaged.
"We felt it big and strong in Honiara, but at the moment here in the hotel all the guests are in the lobby," Samani told Reuters.
Child aid agency Unicef says it is on standby to help and keep children safe.
In 2007 parts of the Solomon Islands were destroyed after a massive tsunami was triggered by a magnitude 8.1 earthquake.
Villages were wiped out and 52 people were killed when devastating waves washed across the island nation.
The largest waves were reported on Simbo Island where waves up to 12m high destroyed about 900 homes and more than 13 coastal villages. - Much of this report is from AAP