Fabian Nabi, aged 27, and Zubear Zaman, 23, were racing each other along Clifton Lane, Clifton, when a 78-year-old man on a mobility scooter was struck in November 2014.
Zaman, of Warwick Street, Moorgate, was jailed for six years after being found guilty of causing death by dangerous driving after a trial. .
Nabi, of Wellgate, Rotherham, was locked up for 40 months after admitting the offence.
Both men were banned from driving for three years.
The men were racing their cars along Clifton Lane, towards Wellgate, when Nabi’s red Audi struck the mobility scooter.
The 78-year-old man riding the scooter was taken to hospital with critical injuries, where he died days later.
Investigating officer Neil Morrell said: “Our condolences are with the family and loved ones of this elderly man, whose tragic death in November 2014 could so easily have been avoided.
“Mr Zaman and Mr Nabi showed no regard for the safety of themselves or others, engaging in a race along a single carriageway road with a 30mph speed limit.
“Their reckless and dangerous behaviour resulted in the death of another person and shattered the lives of his loved ones forever.
“Please let this case serve as a warning that driving dangerously or at speed can have fatal consequences, and that as a police force we will do whatever we can to bring those who drive dangerously before the courts.”