One witness reportedly said the fire ?melted a couple of windows?
Airport officials said at least 13 people were taken to hospital with minor injuries.
US Federal Aviation Administration spokesman Ian Gregor said the left engine of the Boeing 777-200 caught fire before take-off.
The plane was seen engulfed in flames and smoke. The fire was later put out.
The pilot told passengers that the accident was the result of a ?catastrophic failure of the engine?, reports the Guardian?s Jacob Steinberg, who was onboard the plane.
?We looked out the window and for the first time saw smoke near the wing. We could smell it. It was like burning rubber ? bitter and deeply unpleasant ? and it was time to panic,? MSteinberg writes.
Flight 2276 was bound for London?s Gatwick airport.
There were 159 passengers and 13 crew members on board.
The airport tweeted: ?Received first call at 4:13 PM (23:13 GMT); flames were spotted at 4:14, and by 4:14 response was underway.?
It added: [email protected] had the fire out and all passengers off BA flight by 4:18 PM ? excellent work by our ARFF team.?
Airport officials said at least 13 people were taken to the city?s Sunrise hospital, most of them with injuries sustained while sliding down inflatable chutes during the evacuation.
Some of those on board said there was smoke in the cabin as one exit was opened and immediately declared unsafe.
Eyewitness Reggie Bugmuncher said she was waiting at a gate for her flight when she heard people saying: ?Oh, my God.?
She said looked out the window and could see ?bursts of flames coming out of the middle of the plane?, the Associated Press news agency reports.
An investigation into the blaze is now under way after what was clearly a lucky escape, the BBC?s James Cook in Los Angeles reports.
Officials later said one runway was shut down, but three others continued to operate.
McCarran is a major airport in the US, handling more than 40 million passengers last year.