The contract includes $51.1 million guaranteed, a source told ESPN's Ed Werder, the most ever for a cornerback.
Norman's average salary per season will be $15 million, eclipsing Patrick Peterson's contract with the Arizona Cardinals ($14.01 million annually) to make Norman the highest-paid cornerback in the NFL.
"I'm looking forward to getting back to the playoffs here," Norman said in an interview with the team's "Redskins Nation" show on CSN Mid-Atlantic. "You guys went last year. I'm looking forward to going even deeper into the playoffs and then the Super Bowl. That's our motto, that's here, that's the goal. That's the ultimate goal."
Norman stated emphatically that he wanted to win "the granddaddy of them all" with Washington -- and he wasn't referring to the Rose Bowl.
"I want to win the Super Bowl," Norman said. "And I'm not going to stop until I get back there. Because I know I will one day and then make the wrong a right."
Norman, 28, tweeted a series of emoji in the wake of the news, and the Redskins posted several photos, including two of Norman signing his deal.
#HAIL🔥💯💪🏽😾— *Joshua R. Norman (@J_No24) April 22, 2016
Several new teammates congratulated Norman and welcomed him to Washington, including fellow cornerback DeAngelo Hall.
"It came out of nowhere. It really did," Norman told "Redskins Nation" on Friday night. "I got sideswiped. I didn't see it coming."
The Saints had approached Drew Brees about restructuring his contract Friday, but it became moot when they were unable to land Norman, a source told Schefter.
Norman visited the Redskins' facility in Ashburn, Virginia, on Friday. Earlier in the afternoon and Thursday, Washington players had used Twitter to try to persuade Norman to sign. After the agreement was reached, he posed for pictures with family members and Redskins general manager Scot McCloughan.
"When we played you guys last year, y'all was competing all the way until the bitter end. It was no let-up," Norman said in the TV interview. "You guys had the fight in you all. You're a playoff team. It was in the thick of things. And one piece here, one piece there, you guys can do what Carolina [did] when they went to the Super Bowl last year."
Despite the Norman signing, Washington remains 40-1 to win Super Bowl 51 and 20-1 to win the NFC at the Westgate Las Vegas SuperBook.
Norman's regular-season debut will be on ESPN's Monday Night Football on Sept. 12 against the Pittsburgh Steelers. He will face his former team, the Panthers, on Dec. 19, also a Monday night. And joining the NFC East means two matchups against nemesis Odell Beckham Jr. and the New York Giants (Sept. 25, Jan. 1), as well as Dez Bryant and the Dallas Cowboys (Sept. 18, Nov. 24).
Norman had a career-high four interceptions, 16 passes defensed and 56 tackles last season for the NFC champion Panthers.
On Thursday, Carolina general manager Dave Gettleman said the Panthers chose to rescind Norman's tag because of the flexibility the team would have without him counting toward its salary-cap number. Gettleman also said it was unlikely the team would reach a long-term deal with Norman.
"For my old teammates that I had, oh my gosh, I'm gonna miss them dearly," Norman said Friday. "... Being somewhere 28 years all your life and just in a blink of an eye, gone, just like that."
The Panthers had placed the tag on Norman last month, guaranteeing him $13.952 million in 2016. But Norman never signed it and was planning to skip at least the first part of offseason workouts, which begin Monday.
According to a league source, Norman was seeking a long-term deal worth between $15 million and $16 million a year with Carolina, an estimate based on what could be had on the open market. The Panthers made one offer for close to $11 million and never moved off that mark, according to the source.
Information from ESPN's Adam Caplan, David Newton and John Keim, ESPN Stats & Information and The Associated Press was used in this report.