The Saints agreed to terms with the 10-year veteran on Tuesday, with Payton saying at the league meetings in Phoenix that "my understanding is that Jared Cook has signed, and we're excited to have him."
Terms of the deal were not immediately available.
"I think he gives us a threat opposite of Mike [Thomas]," Payton said of Cook, who just had the best two-year stretch of his career with the Oakland Raiders and appeared in his first Pro Bowl in January. "Jared's got a unique skill set and we saw a lot of good tape of him.
"I think he can run. I think he's a really good receiver in space. I think he does a good job with his yards after the catch. He's comfortable playing in a lot of different spots. He's not a great in-line blocker, and yet he is a guy that challenges you if you're playing a linebacker or safety on him."
Added Raiders coach Jon Gruden, who is also at the league meetings: "[It's] sickening to think we lost that guy ... a lot of production we lost there."
The Saints-Cook pairing has been reported as "close" for more than a week. Even Cook himself indicated it was practically a done deal when he told ESPN's Jamison Hensley at an event for the NFL's Ed Block Courage Award winners that "it's going to be pretty awesome" catching passes from Drew Brees.
"I've been a big fan of Drew for a long time and the way he approaches the game," Cook said. "He's on point. He's on top of his game right now. He's just a baller. I look forward to seeing him do great things in the future. ...
"It could be a big production year. Just as long as I take care of what I can take care of, we should be all right."
Cook fills one of the Saints' most glaring needs. They needed another reliable pass-catcher to pair with Thomas and running back Alvin Kamara. Last year, tight end Benjamin Watson ranked third on New Orleans with 35 receptions -- and Watson has since retired.
Cook will turn 32 on April 7, but he is coming off arguably the finest season of his career (a career-high 68 catches, career-high 896 receiving yards and career-high six touchdowns).
Cook spent two years with the Raiders and is the first tight end in NFL history to have a 100-yard-receiving game for four teams: the Titans, Rams, Packers and Raiders.
Last season, Cook had four games with more than 100 yards receiving. He set the franchise record for a tight end with 180 yards -- on nine catches -- in the season opener against the Los Angeles Rams.
Cook was fourth among all NFL tight ends last season in receiving yards, tied for fourth in touchdowns, and was third with 15 receptions of at least 20 yards.
With 5,464 career receiving yards, Cook is the eighth active tight end in the NFL with at least 5,000 yards. He has 425 receptions for an average of 12.9 yards per catch, and he has 25 touchdowns.
He was also the Raiders' winner of the Ed Block award, which honors "those who exemplify commitments to the principles of sportsmanship and courage. The recipient symbolizes professionalism, great strength and dedication."
ESPN's Paul Gutierrez contributed to this report.