Harris was behind the wheel of a new £50,000 Alpine sports car on Monte Carlo's SS17 stage with Jordan in the passenger seat driving up a hill on Monday morning, when its warning lights came on as the engine cut out.
They stopped the car immediately and the presenters made it out safely as flames emerged from underneath the vehicle.
The car burnt through in just four minutes, and Alpine is now investigating the incident.
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Neither Harris nor Jordan, who were both wearing fireproof overalls and gloves at the time, were injured or suffered from smoke inhalation.
Harris said: "I first realised I needed to get out when I opened the door and the flames went up my arm.
"Sadly the car was lost and it always makes me sad to see a beautiful car destroyed."
Jordan said: "Doing a stage of the Monte Carlo Rally was a dream come true for me. The car was stunning - so light on its toes.
"It was dancing around the mountain and Chris was driving it beautifully."
He added: "It's such a shame we didn't finish the test, but these things happen."
A spokesman for the BBC Two show said: "There was an incident during filming in Monaco last Monday morning with an Alpine that led to a fire taking hold of the main vehicle - which Alpine is now investigating.
"Thankfully Chris Harris and passenger Eddie Jordan, both exited the stationary vehicle unhurt."
The motoring show, led by Friends star Matt LeBlanc, will return to screens later this year for its 25th series.