South Yorkshire Police said a 37-year-old man suffered injuries to his wrists, a 12-year-old girl was found unconscious and a 3-year-old girl suffered minor injuries.
At 12.30am on January 1, officers were called to reports of disorder on WIlloughby Street in Page Hall.
A police spokesperson said: "A dispersal order was then granted for the area and dog handlers worked alongside officers to direct people back to their homes.
"We arrested five men, all in their twenties, as a result of the disorder. They have all been bailed pending further enquiries."
The videos of the incident have been circulated widely on social media, showing a heavy presence of police in the area.
Residents claimed 15 police cars and three ambulances were deployed to attend the mass brawl, which had broken out just after midnight.
A new police team was recently set up to tackle issues in the Sheffield suburb such as fly-tipping, fighting, noise and large groups.
Officers have turned an empty house on Robey Street into a police base in a bid to deter trouble and to have staff on hand to respond to incidents and to get to know the community better.
The Page Hall team, which has steadily grown from two officers to around six, was formed earlier this year when large gatherings blighting the area became more of a problem due to the Covid-19 pandemic.
Since the end of October they have seized 50 cars and carried out 17 arrests, along with 101 stop and searches.