Boss signed a $16 million, four-year contract with the Raiders on Friday. The deal includes $8 million in guarantees.
The Raiders, who were looking for a tight end after losing Zach Miller to the Seahawks, announced the deal.
Boss' asking price was clearly too high for the Giants, who have been trying to re-sign him and wide receiver Steve Smith.
"It's disappointing," Giants coach Tom Coughlin said. "We were very much involved. But the decision is the decision. We wish him well."
Boss, who worked out for the Raiders Wednesday, was fifth in receptions last season with 35 catches for 531 yards and five touchdowns. The 6-foot-6 player was one of Eli Manning's most trusted targets and had caught a total of 16 touchdowns the past three seasons.
Boss, who caught 119 passes with 18 touchdowns in four seasons with the Giants, gives the Raiders a 6-foot-6, 235-pound downfield target. He also won a Super Bowl ring with New York in his rookie season.
"He's smooth," said quarterback Jason Campbell, who played against Boss' Giants teams while Campbell was with the Washington Redskins. "He has a lot of experience. We understand he knows what it takes to get to the next level, and what it's going to take for us to get to the next level this year."
The Raiders brought in Boss for a workout Wednesday and signed him after restructuring contracts to get under the salary cap. He did not join the team at its training camp facilities on Friday but coach Hue Jackson expects Boss to be on the field soon.
"I expect to get him here as soon as we can," Jackson said. "We've got work to do. There are quite a few things in our offensive system that a young man needs to do and he showed those characteristics on video tape."
While Boss is the front-runner to replace Miller, the Raiders have holdover Brandon Myers and Kevin Brock on the roster as well. Jackson also lauded the efforts in practice of rookie tight ends David Ausberry and Richard Gordon.
Losing Boss hurts because he provides Manning with a tall safety valve across the middle. Smith, if he re-signs, may not be ready for the season opener due to microfracture knee surgery he had in December 2010.
The Giants could look to free agency to sign another tight end. At the moment in camp, they have Travis Beckum, newly signed former Cardinals tight end Ben Patrick, Jake Ballard and Bear Pascoe, who could play fullback this season.
"It sucks," Beckum said. "Because obviously Kevin's been here since my rookie year, and he's established himself. Kevin and I were close. He taught me a lot of stuff, but it's nothing personal. Kevin knows I wish nothing but the best for him, but he has to do what he has to do, and that's in Oakland."
Beckum will now try to meet high expectations since being drafted out of Wisconsin in the third round in 2009. He had 13 receptions for 116 yards and two touchdowns last season.
"Most of my reps last year were when Kevin was in the game but we are down a tight end," Beckum said. "I'm ready to do whatever I have to do and whatever they ask me to do. I think that I have done a good job with whatever they asked me to do. I took advantage of everything, whether it was coming out of the backfield, at the Y or being in there during goal line. Whatever they asked me to do, I am willing to do it."
Ohm Youngmisuk covers the Giants for ESPNNewYork.com. Information from ESPNNewYork.com's Mike Mazzeo and The Associated Press was used in this report.