The 28-year-old Scott will make $27 million during the first three years of the contract, Lewis told The Associated Press.
Earlier in the day the Jets had offered a five-year, $40 million deal. When New York upped the offer by a year, Scott chose the Jets over Baltimore.
"What made the decision easier for me are the people in the organization and my familiarity with the coaching staff," Scott said in a statement. "These guys really pursued me hard.
"The opportunity to play under coach Ryan again was huge. I've made it no secret how big a role he has played in me taking the steps from being an undrafted rookie free agent to where I am today. I wanted to keep that chemistry going. It's been so successful that I didn't want to lose that magic."
Scott arrived at the Jets' facility early Friday morning to meet with his former Ravens defensive coordinator, Rex Ryan, who is now the head coach of the Jets.
Scott played seven seasons with Baltimore, amassing 433 tackles and 16 sacks in his career. His best season was in 2006, when he made the Pro Bowl and the All-Pro team. In that season, he had 103 tackles and 9½ sacks.
Guard Brandon Moore also agreed to return to New York for a $16 million, four-year deal, a person familiar with the terms said on condition of anonymity because the team hadn't announced the contract.
New York cut Moore on Thursday, but opted to bring him back when Pittsburgh left guard Chris Kemoeatu turned down an offer from the Jets and re-signed with the Steelers.
Scott ranked second on the Ravens in tackles last season and will join promising David Harris as speedy, playmaking linebackers in New York's 3-4 defensive scheme.
"We added The Mad Backer today," Ryan said. "What you've seen him do and accomplish in Baltimore, I think is just the tip of the iceberg. ... He's called The Mad Backer because he basically hates the guys in the other color jerseys."
Scott was a major part of a Ravens defense that ranked No. 2 overall and helped Baltimore reach the AFC championship against Pittsburgh. Scott was believed to be the Jets' No. 1 target this offseason, despite former teammates Ray Lewis and Terrell Suggs -- franchised by Baltimore -- also being free agents.
"I think he's going to be more effective with this Jets defense because we're going to have some flexibility with him and David that, quite honestly, we never really had with Ray," Ryan said. "That's not taking anything away from Ray, but Ray was always the MIKE backer and I think now we can switch Bart and David in and out and it's going to give some identification problems to offenses."
Despite signing Scott, the Jets wouldn't completely rule out also signing Lewis.
"We never thought we'd add Brett Favre last August, so right now we're going to take each day one at a time and look at opportunities," Tannenbaum said. "I would never rule out or rule in any player."
ESPN.com senior writer John Clayton and Andrew Marchand of ESPN 1050 radio in New York and The Associated Press contributed to this report.