Darren Kaye refused to stop when police officers spotted him driving the scooter at speed along Armley Town Street in the early hours of April 28 this year.
Leeds Crown Court heard Kaye ignored a no-entry sign on Branch Road and drove the wrong way down a one-way street.
The 32-year-old defendant went through red lights and refused to pull over when patrol officers put on blue lights and sirens.
He mounted pavements on Crab Lane and almost collided with a car when he went through another red light on Armley Ridge Road.
Carley Green, prosecuting, said officers then lost sight of Kaye on Armley Lodge Road.
The West Yorkshire Police helicopter and a firearms unit were deployed in the search for Kaye.
He was spotted driving at 60pm on Burley Road and was eventually stopped and detained by the firearms unit.
Kaye told officers he failed to stop as he knew he did not have a licence.
The father-of-three said he had borrowed his neighbour’s scooter to go to a 24-hour shop to buy nappies.
Kaye, of St Ann’s Gardens, Burley, pleaded guilty to dangerous driving, driving while disqualified and having no insurance.
The court heard Kaye was given a 12-month driving banning in December last year for accumulating more than 12 points on his licence.
Stephen Smithson, mitigating, said Kaye received a prison sentence for drugs offences in 2015 but had worked hard to stay out of trouble since then.
Mr Smithson said Kaye had moved away from the Seacroft area of Leeds and had been working hard to support his family.
He said: “He was at home and went to his local garage, which was within walking distance, to purchase nappies.
“He could not have been more honest with the police prior to his arrest and interview.
“He pleaded guilty at the first opportunity.”
Judge Tom Bayliss QC told Kaye that an immediate custodial sentence was inevitable.
He was jailed for eight months and banned from driving for 28 months.
The judge said: “It seems that you panicked. In panic you drove in a very dangerous manner.”