Police Scotland said the “very frightening ordeal” began when a man walking in Kilmarnock began damaging cars and struck the blue Mazda 2.
The occupants of the car got into an argument with the suspect, who then entered the car and drove off with it.
The couple tried to stop the car being stolen and were briefly dragged along the street, causing minor injuries.
The robbery happened at around 6.25pm on Wednesday in Low Glencairn Street, at the junction of Maggie’s Bar and Kitchen and Lidl supermarket.
Some 35 minutes later, the car was found on fire at Tarryholme Drive, Irvine.
The suspect is described as white, aged between 20 and 30 with a slim but muscular build.
He has short hair and was wearing a blue jumper.
He is believed to have previously been in the company of a woman in her 20ss with a larger build.
Detective Inspector Andrea Templeton said: “Fortunately neither of the victims were seriously injured, but this was still a very frightening ordeal for them and it was made all the more distressing because of the theft of their vehicle.
“We are pursuing a number of lines of inquiry to identify the suspect and would ask anyone who recognises him, or who witnessed this incident, to contact police immediately.
“Similarly, if your vehicle was in Low Glencairn Street, Kilmarnock, or Tarryholme Drive, Irvine, during the evening of Wednesday January 4 and has dashcam footage that may be able to assist with our investigation, then please also get in touch.”
Information can be passed to police by calling 101, or alternatively an anonymous report can be made to the charity Crimestoppers on 0800 555 111.