A sixth daughter was taken to hospital in a serious condition and an eighth victim was also taken to hospital with minor injuries.
The tragedy is the deadliest in the Scandinavian country since 1998, when a fire in the city of Gothenburg killed 63 people.
The fire broke out around 10.00pm on the ground floor of a block of flats on Kuddbygränd in the district of Rinkeby, about 15 kilometres west of the Swedish capital, and all the victims died of smoke inhalation.
“The fire started in a ground-floor flat and spread through the building, which caused the deaths,” Stockholm police spokesman Jens Wismar told AFP.
Police dismissed reports in Swedish media, which cited witnesses saying the victims died trapped in the building’s lift.
During a press conference on Sunday, emergency services added the bodies were found in the block’s staircases.
The apartment was described by the emergency services as a loft apartment and it is not known how many people were inside when the fire broke out. By the time emergency services arrived on the scene the apartment was ablaze.
Twelve apartments in the building were evacuated and residents were unable to return to their homes later on Saturday night. Temporary accommodation was arranged for those affected.
Many of those interviewed by TT at the scene commented on what they felt was an inordinate delay in the arrival of the emergency services.
“We have called in technicians to examine what has happened,” said Lars Byström of Stockholm police at a press conference in Rinkeby.
TT’s reporter described chaotic scenes in the vicinity of the burning building and on Sunday morning many local residents remained in the dark over the safety of their friends, family and neighbours.
“I was woken up by the sirens during the night, I have friends that live in the building and I am trying to get hold of them now,” a neighbour told TT’s reporter at the scene.
The cause of the fire, which led to the evacuation of around 20 people in the area, was still being investigated but police do not believe it was started intentionally.
Current evidence pointed more towards “dangerous negligence,” police spokeswoman Karin Solberg told Swedish news agency TT.
Built mostly in the 1960s, Rinkeby is a modest district with a large immigrant community.
More than 60 percent of its inhabitants were born outside Sweden, according to figures from local authorities.