UK-born Jones replaced original Crowded House drummer Paul Hester when he left the band in 1994.
News Ltd reports that Jones "succumbed to a battle with brain cancer on Friday".
The message on the Crowded House site states: "We are in mourning today for the death of Peter Jones. We remember him as a warm hearted, funny and talented man, who was a valuable member of Crowded House.
"He played with style and spirit. We salute him and send our love and best thoughts to his family and friends."
Outside of Crowded House, Jones had played with two legendary bands of Melbourne's inner city music scene, Deadstar and Harem Scarem.
A noted session drummer, Jones also played with Rowland S Howard, Edwyn Collins, Stephen Cummings, Mark Seymour, The Wreckery, Lisa Miller and Kate Ceberano.
Crowded House's original drummer Paul Hester died in 2005.