The victims had been travelling in a VW Touran people carrier, when it was hit by a black VW Golf.
Moments earlier, the Golf car was being pursued by police.
Those killed included two men aged 35 and 50 and a woman aged 41. A one-year-old boy died later in hospital.
Three other passengers in the car were also injured.
A 22-year-old woman currently remains in hospital in a critical condition and another woman is serious but stable.
A three-year-old girl is also in hospital with life-threatening injuries.
One man aged 23, and two teenagers aged 17 and 18, who had been in the Golf, were injured. They are under arrest.
The Golf had been travelling along Main Road just after 8.50pm when it collided with the people carrier, which was travelling towards Darnall and preparing to make a right turn.
The Independent Office for Police Conduct (IOPC) has launched an investigation.
An IOPC spokeswoman said there is “no indication” that the police vehicle collided with either of the vehicles involved.
Regional director Miranda Biddle said: “As soon as we were informed, we sent investigators to the scene and we will be carrying out an independent investigation to establish what actions South Yorkshire Police took prior to the crash.
“This is standard procedure when lives are lost following an incident involving the police.
“By attending the scene we have been able to view dash cam footage which confirms that although a pursuit did take place prior to the crash, there is no indication that the police vehicle collided with either of the vehicles involved.
“We have also obtained initial statements from the officers involved and will gather further evidence during the course of our investigation.”
Chief Superintendent Stuart Barton described the crash scene as “horrific”.
He told Sky News: “Last night was a very difficult situation for us to manage but we were definitely supported by members of the community who came out and really helped everyone who was there at the time.
“We know that at around nine o’clock two cars collided – and in one of them was seven people. Tragically, four of those people have died; two males, a female and a one-year-old child.
“We have three people in custody for death by dangerous driving.”
Mr Barton confirmed that there was a “short police pursuit” before the crash but could not disclose any more details amid investigations.
He said the victims killed in the people carrier were “completely innocent people who were out in their vehicle”, adding that the accident was “tragic and awful”.
Sky News’ Tom Parmenter, reporting from the scene, said the family in the people carrier were simply returning home after a trip.
“Unbeknown to them, a police pursuit was going on on the very same road and a Volkswagen Golf which had three men in it was being chased by the police and it is that Golf that collided with the people carrier,” he said.
Dan Jarvis, mayor of the Sheffield City region, tweeted: “Tragic news from #Sheffield today. My thoughts are with the families of those who died, and with the local community in Darnall.”
Detective Chief Inspector Jude Ashmore said: “A high number of resources were deployed to the scene last night, with officers from all of the emergency services working through the night and into this morning to assist.
“Our priority now is to support the victims’ families and those affected, and work hard to determine the exact circumstances around what happened.
“Main Road is expected to remain closed for most of the day as we continue to gather evidence and recover items from the scene.
“I know the local community will be shocked and saddened to hear what happened last night and we will have officers in the area today to speak to residents, should they have any questions or concerns.”
Anyone with information should call 101 quoting incident number 741 of 9 November 2018.