The forward will link up with former Terriers Jermaine Beckford, Joe Murphy and Joe Skarz at the League one side, alongside ex-Town boss Lee Clark.
The deal brings an end to Bunn's three-and-a-half year spell in West Yorkshire, having joined Town from Manchester City in December, 2013.
During that time, he made 98 appearances for Town, netting 17 goals in the process.
The 24-year-old was granted permission to leave the club's Austria training camp earlier this week to finalise his deal at Gigg Lane.
On the move, Town head coach David Wagner said: "Harry is a great character and has been a good player for this club for a long time, but this move to Bury works for everyone now.
"Everyone here at Huddersfield Town would like to thank Harry for what he has done for the club and wishes him all the best for the future."
Bunn has signed a three-year deal with the Shakers.