The 159 passengers and 13 crew members on board got off the plane, McCarran International Airport spokesman Chris Jones said.
Fire officials said 14 people were taken to Sunrise Hospital by early Tuesday evening for minor injuries, most a result of sliding down the inflatable chutes to escape.
The Federal Aviation Administration delayed flights to Las Vegas from some airports for more than two hours after the fire to slow the flow of planes while the disabled Boeing 777 made two of the airport`s four runways inaccessible. One of the runways reopened about 2 1/2 hours after the fire.
Reggie Bugmuncher, of Philadelphia, said she was charging her phone and waiting at a gate for her flight when she heard people saying, "Oh, my God." She looked out the window and could see "bursts of flames coming out of the middle of the plane."
"Everyone ran to the windows and people were standing on their chairs, looking out, holding their breath with their hands over their mouths," Bugmuncher said.
The plane`s emergency slides were deployed a few moments later and passengers quickly got off the plane. She said it was a "bit more orderly" than she would have expected given the dramatic nature of the fire and smoke.
Firefighters stationed at the airport reached the plane two minutes after getting reports of flames, and within another three minutes, everyone inside the plane had escaped.
After firefighters extinguished the flames, emergency vehicles could be seen surrounding the aircraft, which was left a sooty grey from the smoke and fire retardant.
Federal Aviation Administration spokesman Ian Gregor said the plane`s left engine caught fire and an investigation was under way. The National Transportation Safety Board was collecting information about the incident, said Eric Weiss, a spokesman for the agency in Washington.
Clark County Deputy Fire Chief Jon Klassen said the cause of the fire wasn`t clear yet, but the fire didn`t appear to breach the cabin.
One of the airport`s runways was shut down, but operations continued on the other three runways, officials said.
The Boeing 777-200, popular with airlines for its fuel efficiency for long-haul flights, was bound for Gatwick Airport near London.
In its 21-year history, the 777-200 has been involved in two fatal crashes, one in July 2013 that killed three passengers when an Asiana Airlines flight landed short of San Francisco International Airport`s runway, and the Malaysia Airlines flight that disappeared last year and was only recently recovered.
British Airways spokeswoman Caroline Titmuss didn`t answer questions about the incident in an email exchange but said "safety is always our priority."
Titmuss said in an email that the airline was "looking after customers" but didn`t elaborate. She said the airline would release more information later.
Las Vegas` airport is the ninth-busiest in the U.S. and had nearly 43 million passengers last year. The airport has been taking steps to accommodate more international travellers seeking direct flights to Europe and Asia, including adding new gates to accommodate wide-body double-decker jets.
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