Julian Makaa, a spokesman for the National Disaster Management Office, told Australian Broadcasting Corp on Tuesday that 15 deaths had been confirmed in the Solomons.
He said an initial damage assessment was "around 916 houses, and a very rough estimate of the people affected is around 5,000 people".
ABC reported from Papua New Guinea that a family of five had been washed away in the country's far east after a magnitude-8 earthquake off the Solomons' coast on triggered the tsunami on Monday.
Thousands left homeless
The Solomon Islands Red Cross said about 2,000 of Gizo's roughly 7,000 residents were left homeless and that about 500 houses in the town were destroyed, noting that "initial reports from other islands suggest similar or worse levels of damage".
Alex Lokopio, the premier of hardest-hit Western Province, said few of those displaced had even basic supplies, and that their situation would turn desperate within days without help.
Danny Kennedy, a dive shop operator, said people had ventured into town looking for bottled water and other supplies.
"Unfortunately a lot of the stores, their cargo has fallen from the higher shelves and covered lower things, and the buildings are quite unstable," he told New Zealand's National Radio.
Ethnic violence has hindered stability and development independence from Britain in 1978.
A civil conflict a decade later left nation almost bankrupt
Tourism - the islands' key source of income - dealt a blow after post-election riots in June 2006
Of its 500,000 population, three-quarters rely on farming and fishing
According to World Bank the nation is one of the poorest in the Pacific, with average income of $1.60 a day
Marshall said officials would tolerate survivors taking goods until emergency supplies arrived.
"They are desperate times in Gizo," he said. "And we've got to be practical."
One police patrol boat carrying emergency supplies arrived in Gizo from Honiara, the capital, overnight and three private charters were due on Tuesday.
Australian and New Zealand military helicopters in the Solomons as part of a security force there were also expected to take part in relief operations.
The main airport, phone and power lines, and coastal roads remained damaged early on Tuesday.
Manasseh Sogavare, the prime minister, declared a state of emergency and held meetings with his impoverished country's aid donors about getting help.