They were rushed to hospital after a disturbance outside Leeds West Indian Centre, on Laycock Place, Chapeltown, just after 5am on Saturday (March 3).
Police said the victims, two aged 20 and one aged 22, suffered non-life threatening injuries in the attack, which happened after a dispute involving a group of men outside the venue.
A West Yorkshire Police spokesman said: "The suspects were variously described by witnesses as being of Turkish or eastern European appearance or black or mixed race.
"They left the scene in a green Mitsubishi Shogun, registration S862HLG. That vehicle has since been recovered by officers and is undergoing forensic examination."
A 23-year-old man from Leeds was arrested yesterday on suspicion of causing grievous bodily harm with intent. He was later released under investigation, police said.
Detective Inspector Richard Holmes, of Leeds District CID, said: “The three victims received serious stab wounds and the consequences for them could easily have been much worse.
“We are treating this incident very seriously and are carrying out extensive enquiries to identify and arrest those responsible.
“We have already spoken to a number of people who were there but we are still very keen to hear from anyone who witnessed the incident itself or who saw the suspects leaving the scene in the green Mitsubishi Shogun.
“It is possible people may have filmed what occurred on their phones and we would like to hear from anyone who does have any such footage that could assist the investigation.”
Witnesses should call the police on 101 with the crime number 13180102655 or Crimestoppers, anonymously, on 0800 555111.