Two earthquakes measuring between 7.6 and 9 on the Richter scale unleashed a tsunami some 10 meters (30 feet) high that completely inundated the towns of Gizo and Noro. Scores of villagers have been reported missing.
Arnold Moveni, the head of the local emergency relief committee, said rescuers have conducted flights over the hardest-hit areas, and that the death toll will likely keep rising as reports come in from other affected areas.
Solomon Islands authorities have revoked an earlier warning of a possible new tsunami, and have urged local citizens to return to their homes.
There have been no reports of looting so far.
With a population of about 550,000 people the island group is a popular venue with international tourists, and local residents rely mainly on fishing and tourism.
Earlier a spokesman for the Russian Federal Tourism Agency said the chances that Russian tourists were caught up in the disaster were small.
The most powerful and disastrous calamity involving a tsunami over the last 40 years occurred in December 2004, when an earthquake measuring 9 on the Richter scale in the Indian Ocean generated a tsunami that hit South Asian countries, particularly Indonesia, killing more than 280,000 people. Tens of thousands were reported missing.