As many as 27 players also could be sanctioned for their role in the scandal.
Payton said he didn't want the scandal to "taint or tarnish" his team's recent success.
"We'll get through this," he said. "This will be a challenge. ... You know, we've gone through a lot of adversity and we've won a lot of games in really a short window of time. And I know our players are leaders both within the locker room and the coaching staff will look at this as a challenge and a little bit as an opportunity."
Payton said he was confident he will coach the Saints again in 2013, and that his biggest challenge was realizing that for the first time in 39 years, he may not have an active role in football as a player or coach.
"You go through a range of emotions that kind of hit you," Payton said. "You're disappointed. You're disappointed in yourself that it got to this point. I think we're trained as coaches to begin preparation right away. I find myself reflecting on it, and you go through a lot of emotions."
In addition to the penalties for Payton, Loomis and Vitt, Goodell also fined the Saints $500,000 and took away second-round draft choices in 2012 and 2013.
Arthur Blank, owner of NFC South rival Atlanta, praised Goodell's strong punishment.
"I think he dealt with it appropriately," Blank told ESPN.com. "I think it will be one of the most significant decisions he'll ever make as the commissioner. I think he'll be the commissioner for the next 30 years and I think people will look back and say he sent a message to the teams, the players, the coaches, everybody in the NFL and sent a message to the fans that, 'This is not what we're going to have in this league.' "
AP Pro Football Writer Barry Wilner and AP Sports Writer Tim Reynolds contributed to this story from Palm Beach, Fla.
- Already impressing: Utah Jazz rookie Dante...
- Midseason recap and awards for BYU and Utah
- Mired in 3-game losing streak, BYU preparing...
- Utah Jazz give key starters a break in 98-91...
- Booker-ing ram and broken dreams: Midseason...
- Amy Donaldson: Not deciding on a starting...
- Report: Former BYU receiver Cody Hoffman to...
- Utes continue Pac-12 'meat grinder' with...
- Utes top Oregon State in double overtime 114
- Nevada rallies to beat BYU, 42-35 68
- Dick Harmon: BYU amasses yards but... 49
- Mired in 3-game losing streak, BYU... 46
- FBS football coaches vote BYU most... 32
- Peavler: Penalties out of control at BYU 28
- Brad Rock: Utes have their star in... 27
- College football: Utah rises to No. 19... 25