A series of aftershocks, with a magnitude of up to 6.7, kept residents in the area on edge.
Al Jazeera's correspondent, Dan Nolan, said three people were confirmed dead.
He reported that many people living close to the sea in Gizo managed to escape to safety after being warned to flee to higher ground.
Sergeant Godfrey Abiah of the Solomon Islands police said officers in Gizo had been warning residents to move to higher ground away from the coast when the tsunami hit, but communication links were lost soon afterwards.
"We have lost radio contact with the two police stations down there and we're not getting any clear picture from down there," he said.
Nolan said electricity had also been cut off and only a few people were able to communicate by UHF radio or mobile phone.
Wiped out
The quake off the Solomons triggered
tsunami warnings over a wide area
Danny Kennedy, a dive shop owner and provincial politician in Gizo, told Reuters: "There are quite large boats sitting in the middle of the road. Many of the houses that were on stilts are sitting on the ground. A number of the coastal communities have been completely wiped out."
He said the hospital had been evacuated and patients had been taken to a doctor's house on higher ground, with more than 2,000 left homeless.
The National Disaster Management Office said up to 60 buildings had been destroyed including houses.
From the capital Honiara, Solomon Islands Broadcasting Corp. said residents on Simbo island had reported waves travelling up to 200m inland, damaging homes.
Geological agencies, including those in Australia and Japan, put the magnitude of the quake at 8.1 while the US Geological Survey revised its earlier estimate to 8.
The US Geological Survey said the quake struck about 10km beneath the sea floor about 350km northwest of the Solomons capital of Honiara at 7:39am on Monday (20:39 GMT on Sunday).
The Hawaii-based Pacific Tsunami Warning Centre issued a warning for Papua New Guinea and a lower-level "tsunami watch" for other places, including most South Pacific countries.
Beaches in Queensland and New South Wales on Australia's east coast were shut and some ferry services in Sydney were cancelled.
Warren Young, chief lifeguard of Gold Coast City Council in Queensland, said: "We just feel it's best to err on the side of caution in this respect."