The teenager is among a half dozen men in New York and New Jersey whom authorities have arrested since June.
The Russian plane crashed over Egypt's Sinai Peninsula Oct. 31 and killed all 224 people on board.
A homemade explosive device blew up on the plane, Alexander Bortnikov said.
The recording will be analyzed with a technology "unavailable in most countries."
The "chatter" included a boast of taking down the plane on Saturday and how it was done.