They were Terri Harris, 35, her children Lacey and John Bennett, 11 and 13, and Lacey's friend Connie Gent, 11. All three children attended the same school. Oakwood Academy City.
A man has been arrested on suspicion of murder and remains in custody.
Derbyshire Police's Chief Constable Rachel Swann said: "At 7.26am yesterday the force was called in relation to concerns for the safety of a man in Killamarsh.
"Four minutes later, when officers arrived at the house in Chandos Crescent, they found the bodies of 35-year-old Terri Harris, her two children, 11-year-old daughter Lacey Bennett and 13-year-old son John Paul Bennett, and Lacey’s friend Connie Gent, also 11, from Sheffield.
"The families of Terri, Lacey, John Paul and Connie have all been informed and my thoughts, along with all at Derbyshire Constabulary, are with them at this most tragic of times.
"This has been an understandably shocking event that has deeply affected the local community in Killamarsh, as well as Connie’s family and friends in Sheffield, and our thoughts are also with them.
"I would also like to state my thanks to those officers who arrived at the scene with such speed as well as colleagues from the East Midlands Ambulance Service who attended the incident. All acting with immense professionalism in such a difficult situation.
"While understandably worrying and upsetting this is believed to be an isolated incident in which those involved were known to one another and nobody else is being sought in connection with the deaths.
"The force did not have any prior contact with those involved.
"The man officers arrested at the scene, a 31-year-old from Killamarsh, remains in police custody on suspicion of the murders of Terri, Lacey, John Paul and Connie.
"A team of detectives have been working through the night to understand the circumstances that led to their deaths and I would urge anyone with any information to come forward.
"Of particular interest is anyone who was in the Chandos Crescent area between 9.30pm on Saturday evening and 7.30am the following day and who may have information that would be helpful to the investigation."
Lacey and John's father Jason Bennett laid floral tributes at the scene with his parents Trevor and Debbie Bennett today. He had been on holiday when he was informed of their deaths.
Asked about his feelings as left, he said “heartbroken; million pieces; shattered”.
Speaking to Sky News, he said: “I’m broken, I just want to kiss my babies. I just want to kiss them and hold them. TikTok Queen and my beautiful boy, they were beautiful.”
His message to his daughter left with flowers at the scene said: “To Lacey, my TikTok queen.
“Lost without you. Look after your big brother and don’t wind him up. Soon I will be there to kiss you, hold you tight and make it all better.”
His message to his son said: “To John, my beautiful son, keep your kind heart shining while the angels are looking after you for me.”
His mother Mrs Bennett said: “I can’t understand why he had to take them all. Why take the kids? We are just heartbroken.”
She said both children loved football - Lacey was a “proper” girl and John enjoyed his computers.
She said: “I don’t know how he’ll (Jason) come through this. He’s going to need a lot of strength."
Derbyshire Police said they were called to the house at around 7.25am on Sunday and found the bodies of four people.