The Pacific Tsunami Warning Center said a tsunami of about a meter (3 feet) was measured in Lata wharf, in the Solomon Islands. Smaller waves were recorded in Vanuatu and New Caledonia.
The center cancelled earlier warnings for tsunami waves further away.
Richard Dapo, a school principal on an island near Santa Cruz, said he lives inland but has been fielding calls from families on the coast whose homes have been damaged by the waves.
"I try to tell the people living on the coastline, 'Move inland, find a higher place. Make sure to keep away from the sea. Watch out for waves,'" he said.
He said he's heard the waves have swamped some smaller islands, although he's not aware of any deaths or serious injuries at this point. He said it's difficult to contact people because cellphone coverage is patchy in the region.
In Honiara, the warnings had prompted residents to flee for higher ground.
"People are still standing on the hills outside of Honiara just looking out over the water, trying to observe if there is a wave coming in," Herming said. So far, he had received no reports that a wave had been spotted in Honiara.
Atenia Tahu, who works for the Solomon Islands Broadcasting Corp. in Honiara, said most people were remaining calm.
"People around the coast and in the capital are ringing in and trying to get information from us and the National Disaster Office and are slowly moving up to higher ground," Tahu said. "But panic? No, no, no, people are not panicking."
An official at the disaster management office in Vanuatu said there were no reports of damage or injuries there.
More than 50 people were killed and thousands lost their homes in April 2007 when a magnitude 8.1 quake hit the western Solomon Islands, sending waves crashing into coastal villages.
The Solomons comprise more than 200 islands with a population of about 552,000 people. They lie on the "Ring of Fire" an arc of earthquake and volcanic zones that stretches around the Pacific Rim and where about 90 percent of the world's quakes occur.
The U.S. Geological Survey said the quake struck 81 kilometers (50 miles) west of Lata, at a depth of 5.8 kilometers (3.6 miles).
Associated Press writers Nick Perry in Wellington, New Zealand, and Rod McGuirk in Canberra contributed to this report.