Two men, aged 21 and 25, were stabbed after the Volkswagen Golf they were travelling in was involved in a crash with a Mazda.
Violence flared after the cars collided on the roundabout close to Tesco Express, Burncross Road, Chapeltown, at around 7.35pm.
South Yorkshire Police said two men in the Golf were attacked by two men from the Mazda, who are now on the run.
The attackers left the crime scene in a Seat Leon.
Detective Chief Inspector Phil Etheridge, leading the police probe into the incident, said: “Both men were taken to hospital where they were treated for minor stab wounds and have now been discharged from hospital.
“Sheffield CID are continuing with enquiries in the area today to identify those involved and ascertain the exact circumstances surrounding the incident.
“While the investigation is in its early stages, we are treating this as a targeted attack but we need help to understand the reason and who specifically was behind this attack.
“If you were in the area and saw what happened, or have any information, please contact us.”
Anyone with information should call South Yorkshire Police on 101 and quote incident number 767 of April 16.
Crimestoppers can also be contacted, anonymously, on 0800 555111.