Craig Anderson was told he had ''absolutely no regard for the safety of other road users" as he put police and members of the public in danger during two separate incidents.
Anderson, 33, failed to stop for police when they saw him driving dangerously in the Mabgate area of the city on February 27 this year.
He sped off in a VW Golf and reached speeds of 80mph in 30mph areas.
Bashir Ahmed, prosecuting, said Anderson drove straight over a roundabout and drove on pavements to avoid parked cars during the incident at 2.45am.
One of the police vehicles then came across Anderson's car parked up in a side street.
The defendant reversed into the police car and drove off.
A police dog unit van pulled into Anderson's path in a bid to stop him and there was a further collision.
Damage was also caused to a parked car belonging to a member of the public.
Anderson ran from the scene but was caught a short distance away.
A breath test showed that he was more than double the drink drive limit.
Anderson refused to comply with a drug test but told officers he had cocaine in his system.
Mr Ahmed said Anderson claimed not to have been driving the car but had been a passenger.
He was released on police bail pending further investigation but was involved in another police incident in the early hours of April 28.
A police inspector saw Anderson driving an Audi A3 on Easterly Road shortly before 1am.
He again failed to stop and rammed two police cars near to Harehills Lane.
Mr Ahmed said Anderson caused significant damage, leaving one of the cars "in tatters and immovable."
One officer suffered a neck injury and another officer received facial injuries after his airbag was deployed.
The defendant was arrested the next day.
Anderson, of Granville Road, Burmantofts, pleaded guilty to two counts of dangerous driving, driving while disqualified, drink driving, failing to provide a specimen for drug analysis, driving while disqualified, having no insurance and having to licence.
He was jailed for 31 months.
Joanna Shepherd, mitigating, said Anderson pleaded guilty to the offences at an early stage.
She said the offences took place at a time when he was having difficulties in his personal life and was "going off the rails."
Miss Shepherd said: "While it doesn't excuse his behaviour, it does explain why he abused alcohol at that point. That was his coping mechanism."
Jailing Anderson, Judge Tom Bayliss QC said: "I must sentence you for what are grave offences of dangerous driving.
"The facts of the offences are quite shocking
"You have absolutely no regard for the safety of other road users."