The Cardinals sent guard Jonathan Cooper and a second-round pick in this year’s draft to the Patriots in a deal that was pending the two players passing physicals.
The 26-year-old Jones ranked fifth in the NFL last season with a career-best 12 1/2 sacks. Cooper was Arizona’s first-round pick, the seventh selection overall, in the 2013 draft.
He missed his rookie season with a broken leg and never reached the level of play the Cardinals had envisioned for him.
A first-round draft pick out of Syracuse -- the 21st selection overall -- is one of three players who since 2012 have at least 200 tackles, 36 sacks and 10 forced fumbles. The other two are J.J. Watt and Ryan Kerrigan.
The move saves about $5.4 million in cap space for New England and could bolster an offensive line that was battered throughout last season. Two members of its rotation were on injured reserve before Week 6 in 2015, and only once during the regular season did the Patriots start the same offensive line in consecutive games.
It hit a crescendo in in the AFC championship when Brady was sacked four times.
Cardinals general manager Steve Keim had identified improving the pass rush off the edge as his No. 1 priority for team that expects to contend again for a Super Bowl berth.
He seems to have what he wanted.
Jones played in 55 games in four seasons for New England, 52 as a starter, with 36 sacks, 217 tackles, 10 forced fumbles and one interception. He played in nine post-season games with two sacks and 18 tackles.
"Want to thank the â†•Patriots, the Kraft family, and all of Pats Nation for making my time in New England so special," Jones said on his Twitter account Tuesday.
Dwight Freeney, 35 at the time, joined the Cardinals five games into last season and led the team in sacks with eight.
Two of the leaders of the Arizona defence, defensive tackle Calais Campbell and cornerback Patrick Peterson, sent tweets expressing elation with the trade.
Jones, 6-foot-5 and 265 pounds, is in the fifth and final year of his contract although the Cardinals would have the option of placing the franchise tag on him next season if they don't sign him to an extension.
Jones was involved in a bizarre incident last Jan. 10, showing up disoriented at the back door of a police station about 7:40 a.m., dropping to his knees and putting his hands behind his head.
He was not arrested or disciplined by the team, and later apologized for what he called "a pretty stupid mistake."
Cooper started the first 10 games last season and lost his job to Ted Larsen. When asked why Cooper, when healthy, wasn't back in the lineup, coach Bruce Arians said simply that Larsen was playing better.
The trade leaves Arizona with more uncertainty on the right side of its line since Larsen is gone. Last year's starting right tackle Bobby Massie has signed a free agent contract with Chicago. Arians has said he expects D.J. Humphries, Arizona's first-round draft pick in 2015, to move into that spot. Humphries did not play at all last season.
The Cardinals also will have a new centre since Lyle Sendlein will not be re-signed. A.Q. Shipley is the heir apparent if one isn't obtained elsewhere.