The trio were visiting Longleat Safari Park in Wiltshire when the vehicle overheated yesterday afternoon.
Photographs posted on Twitter show the 4x4-type car engulfed in flames in the enclosure, which is home to 12 lions.
A spokesman for Longleat said the three occupants were transferred to a ranger's vehicle before the car burst into flames. No-one was hurt.
Rangers managed to clear the animals from the enclosure and the park was temporarily closed while firefighters extinguished the blaze.
It is understood the woman alerted staff by sounding the car's horn.
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."