The 32-year-old Republic of Ireland international has been deemed surplus to requirements by Bolton manager Dougie Freedman - despite 27 appearances for Wanderers last season.
Brighton boss Oscar Garcia said of Andrews: "He is a player with experience at the top level of the English game and international football - including World Cups and European Championships.
"Keith is a player who I know will enjoy the way we like to play. He is a dynamic and energetic player."
Andrews has previously played for Hull, MK Dons, Blackburn, Ipswich and West Bromwich Albion.