The 31-year-old Gaborik has struggled this season, with just nine goals and 10 assists in 35 games.
The Slovakian right wing scored 41 goals last season, but managed just five in 20 playoff contests as he dealt with a shoulder injury that required surgery in the offseason.
Gaborik has 333 goals and 333 assists in 757 games over 12 seasons with New York and, previously, Minnesota.
He has one more year on a contract that will play $7.5 million US, with a cap hit of the same amount.
New York receives former first round picks Derick Brassard and John Moore, as well as forward Derek Dorsett.
Centre Brassard has seven goals and 11 assists on the season. Right wing Dorsett could be out for the season with a broken clavicle, while Moore has one assist in 17 games.
The deal represents the first major transaction completed by Jarmo Kekalainen, Blue Jackets general manager.
Kekalainen's predecessor Scott Howson worked on a trade last summer with New York counterpart Glen Sather which saw longtime Columbus force Rick Nash head to Broadway, with the likes of Artem Anisimov and Brandon Dubinksy heading the other way.