The deal was announced on Sunday.
Beck, 24, had eight goals and eight assists in 62 games with Nashville last season. It was his first full year in the NHL after spending time with the American Hockey League's Milwaukee Admirals.
Nashville received forward Jamie Devane in the trade. Devane, 24, has spent the majority of his professional career with the AHL's Toronto Marlies, collecting six goals and 253 penalty minutes in 118 games.