Helen Clements was visiting Longleat Safari Park in Wiltshire with her nine-year-old son George and daughter Charlie, 12, on Good Friday when their vehicle overheated.
Photographs posted on Twitter showed the 4x4-type car engulfed in flames in the enclosure, which is home to 12 lions.
Ms Clements told the BBC: "I thought the car had overheated. Then basically, we thought: 'No, that's not steam, that's actually smoke'.
"It was getting thicker and thicker and obviously coming into us, and then obviously we saw flames."
Ms Clements, 43, from Kingswood, Gloucestershire, sounded the car's horn to alert rangers, who managed to rescue the family and clear the animals from the enclosure.
She added: "Unfortunately they were shouting to us: 'Get back in the car, do not get out of the car'.
"It was a situation of - what do you do? Do you get out of the car because you're on fire? And they're telling you to get back in the car, so that was the dilemma."
A spokesman for Longleat Safari Park said the occupants of the vehicle were quickly picked up by rangers who transferred them into their vehicle and the fire service was called. "The car subsequently caught fire after the guests had been escorted out of their car."