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World Vision said at least 100 homes had been destroyed and Government officials confirmed five people had died.
"I am currently walking through one community [in Lata], and I'm knee-deep in water," said Jeremiah Tabua, World Vision's emergency response co-ordinator in the Solomons. "I can see a number of houses that have been swept away by the surge."
The tsunami was triggered by a magnitude 8 earthquake that also caused panic in other Pacific countries and led to Civil Defence placing New Zealand and other nations on tsunami watch, a step below a full alert.
New Zealand Prime Minister John Key said the Government was working with Solomon Island authorities to see what support New Zealand could give.
Foreign Minister Murray McCully had been in touch with the Solomon Islands Government and "made it clear we're here to help if we can".
The hardest-hit areas were four villages in Temotu province in the Santa Cruz group where waves of up to 1m swept ashore and reportedly damaged dozens of homes and properties.
Jared Berends, 35, a New Zealander with World Vision in the Solomons, told the New Zealand Herald panicked villagers ran for their lives when they saw the water coming, which they described as a "tidal surge".
A World Vision staff member was speaking to her sister on the phone when the tsunami hit.
"She ran down the two-storey office and as she did so could see the water starting to come up ...
"Our staff were literally talking to her as she said, 'tsunami', put the phone down and ran out the door," Mr Berends said.
Earthquakes had shaken Lata, the capital of Temotu province and the largest town on the island, all week so there was some awareness of the possibility of a tsunami.
"Most of the communities had been feeling them so there was high alert and some had already gone to higher ground by that point as well - but there was still a mad rush to get to higher ground."
Communication with the affected areas was difficult. "We've had reports that many homes there have been completely destroyed."
There were unconfirmed reports of children being swept away.
Mr Berends said people on Santa Cruz were fearful of the aftershocks but believed the immediate tsunami threat had passed.
Kirsty Taylor-Doig of the New Zealand Defence Force confirmed eight staff in the Solomon Islands were unharmed.
In April 2007, more than 50 people were killed and thousands lost their homes when a magnitude 8.1 quake hit the western Solomon Islands, sending waves crashing into coastal villages.
The Solomons comprise more than 200 islands and have a population of about 552,000 people.