It happened in Westburn Road, Cambuslang, and the youths stabbed were aged 15, 16 and 17.
Emergency services were called and the youngsters were treated at the scene.
They were then taken by ambulance to Hairmyres Hospital, East Kilbride.
The 16-year-old has since been discharged from hospital, but the other two are still receiving medical attention and are said to be "in a stable condition".
A police spokesman said: "We are looking for information about the incident, which took place about 8pm on Friday.
"It was a large scale disturbance and we at are the early stages of the investigation.
"Anyone who saw or heard anything is asked to call us."
Last August more than 100 anti-knife crime campaigners marched through Cambuslang to commemorate a man who was killed in an unprovoked attack.
George Mathieson, 22, a tree surgeon, was stabbed in November 2010 by 20-year-old Darren Smith, a knife offender who was on early release from a young Offenders' Institution.
Smith, who already had four previous convictions for carrying blades, was later sentenced to 19 years for the murder.
Police stopped traffic for the demonstrators as children, families and young people marched through the town to a rally.
Dressed in campaign T-shirts, the marchers waved banners and placards with anti-knife slogans.
Anyone with information relating to last Friday's incident should call police in Rutherglen on 0141 532 5200 or Crimestoppers on 0800 555111.