The Raiders announced Tuck's signing Thursday afternoon.
The Giants' offer fell far below what Oakland offered and Tuck was in contact with Giants higher-ups Thursday. He had hoped to remain in New York, but the Raiders simply wanted him more. Oakland needed for a pass-rusher after Lamarr Houston agreed to sign with the Chicago Bears.
Tuck ranks sixth with the Giants and 99th in NFL history with 60.5 sacks. He had 11 sacks in 2013 after two injury-plagued years in which he totaled only nine. Last season was his fourth with 10 or more sacks. He also won Super Bowl rings as a member of the Giants following the 2007 and 2011 seasons.
In addition to his pass-rushing numbers, Tuck also has a reputation as a strong run-defender and has often moved inside to play defensive tackle in the "NASCAR" package the Giants use when they want to load up on pass-rushers on obvious passing downs.
Tuck will turn 31 on March 29. He was a defensive captain the past several seasons.
ESPN.com Giants reporter Dan Graziano contributed to this report.