The team confirmed the news on Twitter.
Vick wrapped up the deal -- rumored for weeks -- after a Friday visit with the team. He was widely regarded as the top free-agent quarterback, but he didn't generate a lot of interest on the open market. In fact, the Jets were his first visit.
After finishing last season on the Philadelphia Eagles' bench, Vick said recently he wants to be a starter again, so he obviously feels he will have that shot in New York. He's expected to battle incumbent Geno Smith in an open competition. Despite an encouraging finish, Smith hasn't received a commitment as the 2014 starter.
"Make no mistake about it, Geno Smith is going to be hard to beat out, no matter who we add," coach Rex Ryan said Thursday in an interview.
Vick's arrival allowed the Jets to part ways with Mark Sanchez, who was released Friday. A $2 million roster bonus was due Tuesday. Sanchez missed last season, recovering from shoulder surgery. His cap charge is a prohibitive $13.1 million.
Vick will be reunited with coordinator Marty Mornhinweg, who coached him from 2009 to 2012 in Philadelphia. Vick enjoyed his best all-around season in 2010. His familiarity with Mornhinweg and his system played an important role in his decision.
"He's here in a role where he's going to compete and push Geno Smith to become the very best that he can be," Mornhinweg said.
Could DeSean Jackson be joining them? The Eagles are said to be shopping their top wide receiver, and the receiver-needy Jets reportedly made an inquiry.
A Jets offseason wouldn't be complete without a significant move at quarterback. They traded for Tim Tebow in 2012 and drafted Smith last year in the second round.
Sanchez was on the verge of claiming the starting job last preseason when he hurt his shoulder. Smith got the job by default, but there was no experienced backup -- a void that may have cost the team a playoff spot. Ryan stuck with Smith through a brutal slump, in part, because there was no seasoned alternative.
Under pressure after three straight non-playoff seasons, Ryan wanted to give Smith insurance. He has that in Vick, who also could serve as a mentor to the young quarterback.
Vick, who will turn 34 in June, won the Eagles' starting quarterback job in a preseason competition with Nick Foles last summer. After pulling a hamstring in an Oct. 6 game against the New York Giants, Vick tried to return to the field three weeks later. He aggravated the injury and spent the rest of the season as Foles' backup.
The Eagles signed Vick in 2009, after he served his 21-month federal prison sentence and was reinstated by NFL commissioner Roger Goodell. The move was controversial among fans who objected to Vick's dogfighting activities and also because the Eagles already had Donovan McNabb and Kevin Kolb on their roster.
Vick played sparingly in 2009. In 2010, after the Eagles had traded McNabb to Washington, Vick took over the starting job when Kolb was sidelined by a concussion. Vick played some of the best football of his career that season, leading the Eagles to a playoff berth. He was rewarded with a long-term contract, but was plagued by injuries (he missed nine of 32 games) and interceptions (he threw 24) in 2011 and '12.
In his career, including his six seasons in Atlanta, Vick has played in all 16 games only once, in 2006.
Information from ESPN.com Eagles reporter Phil Sheridan was used in this report.