Former soldier Tyrell Marks was jailed for eight months after a court heard he was high on cocaine and cannabis at the time of the collision near to St James's Hospital.
Marks, 25, drove at double the speed limit, mounted pavements, drove across pedestrian areas and ignored red lights.
The chase ended when Marks ploughed into an Audi driven by an off-duty doctor.
Marks abandoned his passenger, who suffered an injury to his arm, and fled.
Leeds Crown Court heard the doctor remained at the scene and gave first aid to the injured passenger.
A judge commended the actions of the doctor, saying: "She is to be praised for the presence of mind and assistance she offered at the scene."
Bashir Ahmed, prosecuting, said Marks refused to stop for police officers when they indicated for him to do so on Ivy Street, East End Park, on August 14 last year.
Marks drove his Ford Ka at 60mph in a 30mph area along Torre Road and went onto the wrong side of the carriageway.
He failed to give way at junctions and went through red lights as he headed toward Leeds city centre.
Marks then headed along Stoney Rock Lane and mounted a pavement on Shakespeare Avenue before driving over a pedestrian area.
He drove across a grassed area on Beckett Street and narrowly avoided hitting a pedestrian on Cromwell Mount.
Marks ran off after colliding with the doctor's car but was arrested on Regent Street after being chased on foot by police officers.
Marks, of Primrose Circus, Little London, Leeds, tested positive for cocaine and cannabis after the incident.
He pleaded guilty to dangerous driving and two offences of drug driving.
Andrew Foley, mitigating, said Marks had spent four years in the army until 2014 and had served in Afghanistan and Cyprus.
Mr Foley said Marks panicked and drove off after seeing police as he had smoked cannabis earlier in the day.
The lawyer said Marks was unaware that he had cocaine in his system at the time.
Mr Foley added: "He realises that it was a stupid mistake and he did not realise the full seriousness of the offence."
Recorder David Osborne told Marks: "The fact remains that the risk to the public in this case was very grave indeed.
"The only thing standing between you and an eight or ten year sentence is good luck that you did not kill or injure anyone by what you did."