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 said: "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."
Ms Clements said her son got out of the car and ran before they were rescued but she called him back.
A spokesman for Longleat said the family were transferred to a ranger's vehicle and no-one was hurt.
The park was temporarily closed while firefighters extinguished the blaze.
Anna Bing, who was visiting the park, said: "We were stuck for 20 minutes in the enclosure not moving. We saw the lions being chased back to their pen by the keepers in a Land Rover quite rapidly.
"Then the queue started moving and we could see smoke. Then the firefighters arrived and as we passed the fire was out."
A spokesman for Longleat Safari Park said: "On Friday April 18 at around 1pm, a car overheated in the first lion enclosure at Longleat Safari & Adventure Park.
"The occupants of the vehicle, a female and two children, 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.
"The lions were cleared from the enclosure and the safari park was closed. A route was cleared for the fire engine, which arrived quickly and extinguished the fire.
"In total the safari park was closed for 15 minutes.
"No-one was injured in any way and no lions were hurt.
"The vehicle has now been safely removed and the safari park has fully reopened to visitors."