The quake struck at 7:14am (2014 GMT Saturday) waking residents in the Solomons capital Honiara about 300 kilometres from the epicentre, and was followed 10 minutes later by a 5.
The US Geological Survey, which measured the undersea quake at around 29 kilometres (18 miles) deep, issued a “green alert”, indicating there was a low likelihood of casualties and damage.
Honiara resident Dorothy Wickham said the National Disaster Council was warning people to stay away from low-lying areas as the islands experienced high waves.
“People are moving away from the coasts and are going up into the hills, but I have not heard of any damage,” she said.
The Intergovernmental Oceanographic Commission initially put out a tsunami warning for the Solomon Islands, Vanuatu, and Papua New Guinea, but has since cancelled the alert.
A warning of a potential tsunami to hit New Zealand, 3,400 kilometres to the southeast, was also withdrawn.
The Solomons were hit by flash floods 10 days ago which left more than 20 dead.
Several more are still missing in Honiara after the city’s main river burst its banks following days of heavy rain, creating a torrent of water that swept away entire communities.
The Solomons are part of the “Ring of Fire”, a zone of tectonic activity around the Pacific that is subject to frequent earthquakes and volcanic eruptions.
A 6.1-magnitude tremor hit the Solomons on Saturday and a powerful 7.1-magnitude earthquake and 6.7 aftershock struck off Papua New Guinea’s Bougainville island on Friday, to the west of the Solomons.
In February last year, the Solomon islands were hit by a major 8.0 magnitude quake that generated small but deadly tsunami waves which washed away houses and reached as far away as Japan.
In 2007, a tsunami following an 8.0-magnitude earthquake killed at least 52 people in the Solomons and left thousands homeless.
The quake lifted an entire island and pushed out its shoreline by dozens of metres.