The quake, with a magnitude 8.0, struck at 01:12 GMT near the Santa Cruz islands, the US Geological Survey said.
A tsunami measuring 0.9m (3ft) then hit Lata on eastern Santa Cruz island, swamping the island's airport.
A tsunami warning was triggered for several Pacific nations, but was cancelled about two hours later.
Lata is the main town on Santa Cruz, also known as Nendo. It is the largest island in the Santa Cruz island chain, part of the Solomon Islands nation.
The worst of the damage was said to have been on the western coast of Santa Cruz, with one report putting the waves there at 1.5m.
Medical staff at Lata hospital said five people had been killed - four elderly people and one young boy.
Watch from Reuters: