Shabbina Begum, 54, and her 20-year-old daughter Anum Parwaiz Kayani, are also believed to have died in the blaze that broke out in Wake Road, Sharrow, Sheffield.
The cousin of the 54-year-old, Ishfaq Hussain Kayani, said: "It's just a huge tragedy. Words cannot describe how sad this is.
"It's a tragedy beyond description."
He said a nine-week-old baby girl was among those who had died, as well as nine-year-old Adhyan Parwaiz Kayani and seven-year-old Amaan Parwaiz Kayani.
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 of Ms Begum: "She was, to my children, like a mother, like an aunt.
"The whole community is in huge shock.
"As soon as the news broke just after midnight, we came round. I've been with the family.
"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. We are a close-knit family, there are enough people around to support the family."
He added: "They were such a lovely family, highly respected.
"Quite an industrious family who had everything going. This tragedy has struck and we have to try our level best to pull through this."
He described how their mother, Razia Nazim Kayani, survived the blaze but was now in a state of shock and "beyond consoling".
"I believe the mother of the children, I think she was in the house," he said.