Passenger Ian Smith retold the events of the flight to ABC 6, and said the plane was 30 minutes out when it suddenly fell through the sky.
He said: "Thirty minutes out. They were giving us our drinks. The flight attendants were in the last couple rows when they said 'fasten your seat belts.'
"And then they said for the flight attendants to get to their seats, and they didn't even have time.
"It started shaking, then it took a big drop. Babies screaming, people in front of us hitting the ceiling."
ProPublica reporter Jessica Huseman was another flier who went through the terrifying experience.
She wrote on Twitter: "No warning at all. Plane lurched thru the air. Honestly, terrifying.
"Turbulence on flight was so insane and unexpected a flight attendant dislocated his shoulder."
Huseman added: "Some passengers are injured as well. We all used our fleece blankets to mop the ceiling of the plane down after drinks went flying.
"They'd *just* handed out drinks. This plane (that I'm still on) is soaked. Soaked. They'll be hosing this out for hours."
The flight landed at 3.10pm but it was originally expected to land at 3.45pm, according to the airline's flight arrival times.
There were 287 passengers and 12 crew members on the flight coming from Athens.
The injured were taken to a nearby hospital for evaluation. There was no immediate word on their injuries or whether any would be admitted.
American said in a statement: "We are taking care of our passengers and our crew members at this time and want to thank our team members for keeping our passengers safe."