Sajid Majeed and Mohammed Safraz were found guilty of dangerous driving and criminal damage over the incident in Harehills.
Leeds Crown Court heard the crash happened on May 10, 2020, after both defendants had been to Harehills Cemetery.
John Hobley, prosecuting, said Majeed was driving an Audi and Safraz was in a VW Golf.
The prosecutor said CCTV footage captured both vehicles driving in the opposite direction to each other shortly before the incident.
After passing each other Sarfraz stopped and reversed his car into a side road.
Majeed continued in the same direction for a short distance before turning around and driving back towards the direction of where Sarfraz had parked.
Moments later Sarfraz pulled out of the side road and rammed Majeed's vehicle.
Mr Hobley said Majeed's car spun around so the defendants were side by side, with Sarfraz's car on the wrong side of the road.
The prosecutor said Majeed then "nudged" his rival's car and caused it to hit another car travelling along the road.
The collision caused £1,900 worth of damage to the other vehicle, which was not connected to the dispute between the two defendants.
Both men were arrested after the incident and each denied responsibility for causing the damage.
Sarfraz, 34, of Sandhurst Avenue, Harehill, and Majeed, 27, of Montagu Drive, Oakwood, were both found guilty of dangerous driving and criminal damage after a trial.
Mark Foley, mitigating for Majeed, said his client accepted the verdict of the jury.
He described Majeed's actions as "an instinctive impetuous thing".
Mr Foley said: "There is a very regrettable family dispute, without which this dispute would not have occurred.
"There are two sides to any argument and it is accepted that both defendants would put a different explanation as to the background of it."
Timothy Jacobs, for Sarfraz, said his client had held a driving licence for 17 years without being in trouble before.
Mr Jacobs said: "He is ashamed and regretful of what happened two years ago.
"He is a respectable man who does a great deal for his family and other people."
Majeed was given a three-month prison sentence, suspended for 12 months.
He was also ordered to do 100 hours of unpaid work and pay £975 compensation to the driver of the other car.
Safraz was given a three-month sentence, suspended for 12 months and was made the subject of a curfew order for four months.
He was also told to pay £975 compensation.
Recorder Anesh Pema said: "Both of you acted in a way which the jury found dangerous.
"This was in the middle of the day. Other cars were in the vicinity. One of those cars was struck and damage was caused.
"What if it had been a child that was struck?
"Your joint actions caused that collision."