The quake struck 211 miles east of Kira Kira in the Solomons, the Pacific Tsunami Warning Centre in Hawaii said as it issued warnings for South Pacific nations including Australia and New Zealand.
A tsunami measuring 3ft hit near the town of Lata on the remote Santa Cruz island, swamping villages and the airport as people fled to safety on higher ground.
More than three dozen aftershocks, up to magnitude 6.6, rocked the region in the hours after the quake, the US Geo- logical Survey said.
Lata hospital's director of nursing, Augustine Pilve, told New Zealand television five people had been killed, including a boy about 10 years old. He said more casualties were possible as officials made their way to at least three villages that may have been hit.
"It's more likely other villages along the coast of Santa Cruz may be affected."
Disaster officials in the Solomon Islands capital Honiara said they believed six people had died and five villages had suffered damage.
Solomon Islands Police Commissioner John Lansley said it was too early to fully assess the damage or casualty numbers, and authorities hoped to send aircraft to the region to help determine the extent of the damage.
Luke Taula, a fisheries officer in Lata, said he watched the tsunami, which came in small tidal surges rather than as one large wave.
He said: "We saw small waves come in, then go out again, then come back in. The waves reached the airport terminal."
The worst damage was to villages on the western side of a point that protects the main township, he said.
"There are reports some communities have been badly hit, their houses have been damaged by the waves."
About 5000 people lived in and around the town, but the area was deserted as people fled to higher ground, Mr Taula said.
The Solomons, perched on the geologically active Pacific Ring of Fire, were hit by a devastating tsunami following an 8.1-magnitude quake in 2007.
At least 50 people were killed then and dozens left missing and more than 13 villages destroyed.
"It's an area that is very prone to earthquakes," said Jonathan Bathgate, seismologist at Geo-science Australia.
"We've had seven 6-plus magnitude earthquakes in this region since January 31, so it has been very active in the past week."
Authorities in the Solomons, Fiji, Guam and elsewhere had urged residents to higher ground before the Pacific Tsunami Warning Centre cancelled its alerts.
"The earthquake would have to be quite a bit bigger to make a much more sizeable tsunami," said Brian Shiro, geophysicist for the centre in Hawaii.