Shabbina Begum, 54, and her daughter Anum Parwaiz Kayani, 20, are also believed to have died in the blaze that broke out in Wake Road, Sharrow, Sheffield.
Mrs Begum’s cousin, Ishfaq Hussain Kayani, said: “It’s just a huge tragedy. Words cannot describe how sad this is.”
He said a nine-week-old baby girl was among those who had died, as well as Adhyan Parwaiz Kayani, nine, and Amaan Parwaiz Kayani, seven. Ms Parwaiz Kayani was the children’s aunt.
Mr Kayani said Mrs Begum had escaped from the house, but had gone back in to try to rescue the children and had subsequently died.
Describing the children as “full of life, quite bubbly and very energetic”, he said they had been looking forward to returning to school.
He said: “The whole community is in huge shock. It’s so sad. A couple of hours ago the whole family was leading their normal life, quite prosperous, happy.
“Just in a matter of minutes, the whole family has been destroyed. The whole family has gone. You can well imagine what the children’s parents are going through. I don’t have the words to describe how the family is feeling, other than shell-shocked, devastated.”
He said the children’s mother, Razia Nazim Kayani, survived the fire but was now in a state of shock and “beyond consoling”.
He said: “She [the mother] managed to escape. The grandmother of the children, she made attempts to rescue the children, but obviously it wasn’t to be.
“She came out then went back in, tried her level best. Frantic attempts were made to rescue the children.”
He said the children’s father, Nazim Parwaiz Kayani, who is a taxi driver, was out at work when the fire broke out.
Locals said dozens of people came out in the street to try to offer help as the emergency services arrived. Some neighbours reported hearing screams and crying from the house when they were woken in the early hours of the morning.
South Yorkshire Fire and Rescue said the fire began shortly after midnight on the residential street.
A fire service spokesman said: “An investigation has commenced, involving police and fire officers, to establish the cause of the fire.”
Four fire engines were called to the scene and an aerial ladder was used to tackle the blaze.
Police and ambulance service personnel also went to the house after receiving emergency calls and a cordon.
A South Yorkshire Police spokesman said: “At this stage, we are not able to confirm where in the house the fire broke out, or if anyone else was injured.”
Phil Shillito, South Yorkshire Fire and Rescue area manager, said 25 firefighters were sent to tackle the blaze.
He said: “This is a tragic reminder of the devastating effects of fire and our thoughts are with the families of those involved in this incident.”
Local councillor Qubran Hussain, who knew the family, said he believed smoke alarms alerted people in the street, who broke windows to try to get into the property.
Mr Hussain said that he came down to the scene in the night.
He said: “The lady came out – the grandmother came out. But then she found all the children were inside so she went back in and she’s also died.”
Mr Hussain said he employed Mrs Begum’s husband for many years as a taxi driver in nearby Dronfield. He said he thought the children’s father was out working at the time.
The whole of Wake Street and parts of neighbouring streets were cordoned off yesterday and guarded by a large number of police. Used ambulance equipment was strewn on the pavement near the terraced house.