Palace have paid a reported B£2million for the 22-year-old right-back, who has signed a four-year deal.
"I'd like to thank the chairman Dean Hoyle and the board of directors for everything during my time at the club," said Hunt, who joined Huddersfield's academy as a nine-year-old.
He added on the club's official website: "If it wasn't for them I would have probably moved on after my loans away, but they showed incredible faith and trust in me and have played a massive part in starting my career.
"I would also like to thank all Huddersfield Town fans. As a young player, they really helped me settle into the team - right from my full debut at Arsenal and ever since.
"I've been at the club for a very long time and it will always hold a special place in my heart, but now I have an exciting new challenge to go and play Premier League football against some of the world's best players.
"I've made a lot of friends at Huddersfield Town in terms of players and coaches and I would like to thank them all too.
"With the chairman, Nigel Clibbens, Ross Wilson and the manager Mark Robins in charge, I believe the club's future is very bright."