The elephants attacked two houses on a hill-top late Wednesday in the forest of Banshkhali, about 40 kilometres (25 miles) south of the city of Chittagong, said local police chief Jahirul Islam.
Three women and two children died although male members of the family survived, he said.
Elephants frequently damage property and crops in north-eastern India and Bangladesh where loss of their natural habitat due to population encroachment brings them into conflict with humans.
According to a recent survey, at least 200 people in Bangladesh have been killed by elephants since 1996.
Three-quarters of the attacks took place in the forest areas of the Chittagong hill tracts, with most of the others recorded in the northern hill areas of Mymensingh.
The study found that there were around 800 elephants, including 350 wild ones, left in the country.
Bangladesh, which has a population of 144 million, is one of the world's most densely populated countries with about 1,000 people per square kilometre.

Copyright AFP (Agence France-Presse), 2006