The 82-year-old English magazine is considered one of the world's leading classical-music periodicals, and its annual awards are among the most prestigious in the music world.
Inverne, 30, was previously the European performing arts correspondent for Time magazine and a freelance writer for other publications, including Playbill.com.
Jolly, the The Gramophone's editor since 1990, will become editor in chief. He will continue to contribute to the magazine, as well as overseeing special projects.