The Milton Keynes outfit will receive an undisclosed compensation fee for the 21-year-old, who made 32 League One appearances last season.
"We are delighted that Adam has joined us," Brighton's head of football operations David Burke told the club's official website. "Our scouting department have been aware of him and impressed with his progress at MK Dons.
"He has all the attributes that we are looking for and already has a good knowledge of our playing style, having come through the ranks at MK Dons.
"He has great potential to establish himself in the Championship, and we are delighted that he has chosen Brighton & Hove Albion."
Chicksen was born in Milton Keynes and came through the Dons' youth system.