A grey Ferrari 458 and a red Porsche were involved in a smash on the Tinsley roundabout, close to the M1, in May last year.
The driver of the Porsche, 27-year-old Henry James Hibbs, from Middlefield, New Ollerton, Nottinghamshire, pleaded not guilty to dangerous driving but was convicted after a week-long trial at Sheffield Crown Court.
The driver of the Ferrari, 32-year-old Carl Hartley, from Swadlincote, Derbyshire, fled the crash scene before police officers arrived but later identified himself as one of the drivers involved and claimed his brakes had failed, causing him to collide with the Porsche.
He admitted dangerous driving.
Both men are to be sentenced at a date yet to be fixed.
During the police investigation into the circumstances surrounding the collision, the airbag unit from the Ferrari was tested and an expert established that the car had been travelling at 68mph when it was deployed.
Tests on the car failed to find any evidence of braking system damage.
South Yorkshire Police said that following an appeal for witnesses to come forward, members of the public who had seen the cars earlier in the evening described ‘dangerous,’ ‘unsafe’ and ‘racing style’ behaviour.
Both vehicles were captured on CCTV footage at various locations exceeding the speed limits of the roads they were on.
Acting Sergeant Rodney McEnery, who led the investigation said: “I would like to thank the members of the public who came forward and assisted us with this investigation.
“Your help and commitment to making South Yorkshire a safer place does not go unnoticed.
“Vehicles racing and travelling at speed are a dangerous combination, both drivers were extremely lucky that neither were seriously injured or caused injury to an innocent road user.”