The arrested man is understood to have been the boyfriend of the mother found dead alongside her two children and one of their friends, who had been sleeping over.
The bodies of John Paul Bennett, 13, Lacey Bennett, 11, their mother Terri Harris, 35, and Lacey's friend Connie Gent, 11, were discovered at a property in Chandos Crescent in Killamarsh, near Sheffield, on Sunday morning.
The force added that the man, Damien Bendall, was taken to hospital with self-inflicted injuries.
Derbyshire Police said the murder suspect was detained at the property and had only minor injuries.
The father of two of the children found dead said he is “heartbroken” as police named all four victims.
In a statement released on Monday afternoon, Chief Constable Rachel Swann said: “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.”
Around 300 people gathered on Monday evening for a vigil in a park a few hundred yards from the police cordon, where they placed candles, flowers and teddy bears.Internet Explorer Channel Network