Police were called out to Abbeydale Road, Sharrow, following reports of 'seven or eight men' fighting outside Fades barber shop.
He was taken to hospital for treatment but has since been discharged.
A South Yorkshire Police spokesperson said: "An 18-year-old man and a 16-year-old boy have been arrested in connection to the incident.
"Both were arrested on suspicion of assault and possession of an offensive weapon.
"Enquiries are now ongoing in relation to the incident and anyone with information should call 101 quoting incident number 669 of 12 May 2018."