Galette confirmed that the extension is worth $41.5 million, with $23 million guaranteed -- which is all new money in addition to the two years and $4 million Galette was still due to earn in 2014 and 2015.
Galette could increase that total to $48 million if he reaches 12 sacks this season, according to a league source. He received a $3.5 million signing bonus and is scheduled to receive a $12.5 million roster bonus next year.
"I feel like I'm gonna give 'em a discount," Galette said. "But I'm content with what I got. If you're not happy with what they gave me, then you'll never be content."
Galette also was selected as a captain by his teammates for the first time Monday.
"It feels great, man," he said. "This is the best week of my life."
Galette, 26, began his career as an undrafted free agent out of Stillman College in 2010. The 6-foot-2, 258-pounder had a breakout season with 12 sacks last year in his first season as a full-time starter.
Galette signed a three-year, $7.5 million contract in 2013, when he was a restricted free agent. But the final year would have been voidable if he had reached another 12 sacks this season and played a cumulative total of 60 percent of the team's snaps in 2013 and '14.
Galette said he was happy to get the contract "monkey off my back." And he said he'd love to retire in New Orleans, where he loves the people and feels like they love him back.
But Galette, who described himself as growing up "broke in the slums of Haiti with nothing to eat," said he won't change the way he's always been driven to succeed.
"I definitely wasn't a front-runner. I wasn't born with a silver-spoon in my mouth either. I know what it feels like to have a chip on your shoulder," Galette said. "Keep grinding, and it's going to get there. I believed in that. When people didn't believe in me, I believed in myself. And that's all that mattered."
When asked if he has that silver spoon now, Galette said, "It's a gold spoon now. It's in my son's mouth as well, so I know he's going to be good, he's going to grow up with a different lifestyle than I grew up in. So that's all that matters."