Witnesses reported seeing the suspect run away from the scene on Belle Isle Road in flames, and a video of the incident was apparently circulated on social media app Snapchat.
The car was parked outside its owner's home when it was targeted on Monday night.
West Yorkshire Police said:
"At 10:50pm on Monday police received a report of a silver Vauxhall Corsa having been deliberately set on fire in Belle Isle Road while parked outside the owner’s address.
"Firefighters had attended and put the fire out. They confirmed a window had been smashed and flammable liquid used to ignite the vehicle.
"Witnesses reported seeing the suspect run off with flames on him and he is likely to have received burn injuries.
"Enquiries are ongoing into the incident.
"Anyone with any information that could assist the investigation is asked to contact officers via 101 quoting crime reference 13180490541 or call the independent charity Crimestoppers anonymously on 0800 555 111."