The drama began in the early hours of Tuesday morning when a car failed to stop for a marked police vehicle in Headingley.
During a pursuit, the car, which was carrying two men, travelled on the wrong side of the A58 dual carriageway before colliding with a taxi at Sheepscar Interchange.
Two other police vehicles responding to the incident then collided with each other close to the scene of the first crash, but no officers were hurt.
West Yorkshire Police said:
"At 1.20am yesterday morning a black Vauxhall Corsa failed to stop for a marked police unit.
"A pursuit was authorised and continued to the Sheepscar Interchange where the Corsa travelled alone up the wrong side of the A58 dual carriageway and was involved in a collision with a taxi.
"The two occupants of the Corsa ran from the scene.
"A 24-year-old man was arrested nearby by officers on suspicion of causing serious injury by dangerous driving. He was later further arrested on suspicion of possession of an offensive weapon.
"The taxi driver received minor injuries and the 23-year-old male passenger was treated at hospital for a fracture to his thumb.
"A road closure was put in place while the scene was dealt with and the vehicles recovered. This was lifted at 5.30am. The arrested man was later released pending further enquiries. Enquiries are ongoing to trace the other male from the vehicle."