Town are understood to have received a fee of around £2million for the 22-year-old right-back, who has progressed through the club's academy he joined as a nine-year-old.
Hunt announced on Twitter: "Absolutely delighted to have signed for Crystal Palace. New challenge in my life and can't wait to get started."
Town Manager Mark Robins told the club's official website: "It's a great move for Jack and it's also the right move for the club.
"It's important for us to continue to show this club develops players, has a flourishing academy which will produce more homegrown players for the first team.
"I'm sure our fans will immediately wonder if we will now look at bringing in another player, but Jack's departure doesn't impact upon our recruitment policy and plans at the club.
"As a team we are constantly looking at players and assessing opportunities to improve and take things forward, as we have demonstrated, and that will not change."
Huddersfield handed Hunt his full debut against Arsenal in the FA Cup in January 2011 and he has gone on to make 121 appearances for the club, scoring four goals.