Police officers found the youngster crying on the passenger seat and not wearing a seat belt after the crash in Beeston.
Moments later Nathan Sharp returned to the vehicle but denied being the child's dad or the owner of the car.
Weeks after the incident Sharp subjected the child's mum to violence in her own home and attacked pub door staff.
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Leeds Crown Court heard police officers saw Sharp in a car on Tunstall Road, Beeston, at 8.30pm on February 28 this year.
David Ward, prosecuting, said the officers followed the vehicle as they believed he was a disqualified driver.
Sharp accelerated away from the officers and drove above the speed limit through the Beeston area.
He mounted the pavement when he reached Malvern Road and caused damage to a parked car.
The 27-year-old defendant then drove onto a pedestrian area and onto a footpath before crashing into a fence.
Sharp ran from the vehicle, leaving the child inside.
The officers found the child crying in the front passenger seat of the vehicle.
He was not in a child seat and was not wearing a seatbelt.
Sharp walked back to the car and pretended to be a local resident coming to find out what was happening.
He was arrested when the officers recognised him.
Sharp claimed he had not been the driver and denied being the child's father despite the youngster screaming "Daddy".
He was searched and found in possession of a small amount of cocaine.
Sharp was released on bail but assaulted his former partner and two of her friends on March 25.
Mr Ward said Sharp turned up at his former partner's home with his new girlfriend.
Before turning up at the victim's home Sharp had contacted her and said his girlfriend wanted to fight her.
The two women started fighting and Sharp joined in the violence, punching the woman until she was almost unconscious.
The victim was six months pregnant at the time of the attack.
The woman's two female friends were also attacked by Sharp when they tried to intervene.
Sharp punched one of the women when she was holding a child.
Sharp was involved in further violence the next day, March 26, at the Broadway pub, on Dewsbury Road, Beeston.
Mr Ward said Sharp went to the pub with his new girlfriend.
Staff believed he was under the influence of drugs at the time.
The defendant slapped his girlfriend across the face while they were on the dancefloor and three door security workers tried to intervene.
Sharp punched all three men during a struggle as he was removed from the premises.
During the incident he kept saying "you don't know who I am" and also threw a chair.
Sharp was arrested the next day and told police officers he could not remember being in the pub.
He pleaded guilty to dangerous driving, driving while disqualified, possessing cocaine and six counts of common assault.
Sharp, of St Lukes Green, Beeston, has previous convictions for drug and driving offences.
Stuart Field, mitigating, said Sharp was sorry for the offences and had pleaded guilty at an early stage.
Mr Field said the offences were committed during a traumatic time in his life when his relationship with his former partner had ended.
Sharp was jailed for 16 months and banned from driving for two years.
Describing the driving offence, Deputy District Judge Timothy Clayson said: "A significant aggravating factor is that your three-year-old child was in the front seat unrestrained at the time."
The judge added: "Your record does you no favours whatsoever.
"It is perhaps to be hoped that this present sentence will act as a wake-up call to you.
"You really do need to do something about the way you behave.
"You can't carry on breaking court orders as you wish, nor can you resort to violence in situations which annoy you."