The Jamaica Gleaner reported that three passengers were seriously injured.
Charley Wilson, an American Airlines spokesman, said the aircraft came to a safe stop and there were no immediate reports of fatalities.
Gary Allen, a senior Jamaican journalist, told Al Jazeera it was amazing nobody was killed in the crash.
"It seems as though the remarkable thing is that it overshot the entire airport, crossed a roadway without touching it and then landed on a sandy embankment just on the edge of the beach," he said.
Meanwhile, Darien Henry, a senior reporter who covered the story for Radio Jamaica, said passengers were still awaiting information about the cause of the crash pending the results of an investigation.
"There has been a lot of speculation as to what might have led [to the crash]," he told Al Jazeera.
"One [theory] is that the weather was so unstable and so stormy and that might have caused some difficulty for the pilot. But of course, those are preliminary investigations.
"Information that we have been garnering from passengers suggests that the plane was landing ok and then it all went awry and passengers were strewn all around.
He said some passengers later raised concerns about the response time of emergency crews.
"We have gotten a preliminary response from airport officials and they are defending the actions they have taken saying they responded on time."
The flight was carrying 148 passengers plus six crew members, media reports said.
It took off from Miami airport at 8:52pm (local time) on Tuesday and was scheduled to arrive at Kingston airport at 10:27pm.