The religious sect first rang asking to talk to a woman that, according to their records, had attended a self-realization course in the 1980s.
But it turned out to be case of mistaken identity and the church was duly informed.
However since that initial contact the sect has called the family's phones on more than 70 occasions.
Despite repeated requests to stop harassing them, the calls have continued unabated.
A letter even arrived from the church last Thursday apologizing for the calls, explained the circumstances, and conceded the case of mistaken identity.
But still the calls continued.
The Kävlinge family finally had enough and decided to report the matter to the police.
"If this is true then it is very regrettable. We are busy updating our large database of members in Sweden and there must have been a mix up," a spokesperson for the Church of Scientology in Malmö told The Local on Tuesday.
The Church of Scientology is a new religious movement founded in 1954 and is based on the writings of US science fiction writer L.Ron Hubbard.
The movement has been established in Sweden for 40 years and has been registered as a religious community (trossamfund) since 2000.
In November 2008 the group completed the 50 million kronor ($6.8 million) purchase of sugar firm Danisco's former head office in Malmö suburb Arlöv as their new base in southern Sweden.