In a related move, the Bears released 2016 first-round pick Leonard Floyd, the team announced.
The defensive end's deal with the Bears is for five years and $70 million with $30 million fully guaranteed, NFL Network reported.
Quinn, who turns 30 on May 18, led the Cowboys with 11.5 sacks in 2019, despite missing two games due to suspension for violating the league's policy on performance-enhancing drugs.
He has recorded at least four sacks in each of his nine seasons and reached double figures four times.
The Cowboys acquired Quinn in a trade from the Miami Dolphins last April. His 11.5 sacks in 2019 were his most since notching 19 for the Rams in 2013.
Quinn has 80.5 career sacks and has started all but one game in which he has played in each of the past three seasons.
Without Quinn, the Cowboys' pass rush will need a boost. While they have DeMarcus Lawrence, whose sack total dipped to five in 2019, they do not have a ready-made replacement for Quinn. They could have Tyrone Crawford back after he was limited to just four games before undergoing surgery on both hips. The team also likes Dorance Armstrong's potential.
Quinn's arrival signals the end for Floyd in Chicago. The Bears picked up Floyd's fifth-year option prior to last season, but the team opted to release him before the $13.2 million option became fully guaranteed on the first day of the league year.
Floyd -- the ninth overall pick in 2016 -- started 54 games and notched 18.5 career sacks for the Bears, but he never developed into a consistent pass-rusher, even when paired with perennial Pro Bowler Khalil Mack, who commands double-teams on virtually every snap.
Floyd, 27, recorded a career-low three sacks in 2019.
"I think Leonard wants to be more productive as a pass-rusher," Bears general manager Ryan Pace said at last month's NFL combine. "We want him to be more productive there too. He's close in a lot of areas when you look at the pressures and those things. He just needs to finish a little better on the quarterback.
"But I think when you're evaluating him, you have to factor in everything. His run defense, his ability in coverage. We consider him our Sam outside linebacker, so we value what he can do in coverage and think sometimes that goes a little underrated for what he does in that area, for a guy that's of his stature. Not many outside linebackers can drop in coverage like he does, so that's a factor."
Floyd joins an ever-growing list of disappointing former Chicago first-round picks. The Bears released receiver Kevin White -- the seventh pick of the 2015 NFL draft -- after four injury-plagued seasons. The team is also actively attempting to acquire a veteran quarterback to compete with Mitchell Trubisky, one of the league's lowest-rated passers. Chicago traded up to draft Trubisky second overall in 2017.
ESPN's Todd Archer and Jeff Dickerson contributed to this report.