The Rangers also announced the firing of assistant coaches David Oliver, Jacques Martin and Greg Brown.
Quinn went 96-87-25 in his three years with the Rangers, including a 27-23-6 record this season, which left New York 11 points shy of a playoff spot.
The Rangers hired Quinn in 2018 after missing the playoffs in 2017-18 for the first time since the 2009-10 campaign.
New York parted ways with the accomplished Alain Vigneault in order to hire Quinn, but it was a move that ultimately didn't pay dividends.
The Rangers went 32-36-14 in Quinn's first season at the helm before going 37-28-5 last season. That would not have been good enough to get the Rangers in the playoffs under normal circumstances, but they did technically compete in the postseason.
Due to the COVID-19 pandemic cutting the 2019-20 season short, the top 12 teams in the Eastern Conference were part of the playoffs. New York's playoff run was short-lived, though, as it got swept in the qualifying round in three games by the Carolina Hurricanes.
The Rangers were in the playoff hunt for much of this season, but playing in the stacked East Division, they could only manage a fifth-place finish.
Quinn, 54, coached at multiple levels of professional and collegiate hockey before taking the head job with the Rangers.
Most notably, he was the head coach at Boston University from 2013-18, leading the Terriers to the NCAA tournament on four occasions, including a trip to the national championship game in 2015, which they lost to Providence.
The coaching change was expected with Chris Drury taking over as general manager, and whomever he picks to succeed Quinn will be inheriting a talented roster featuring forwards Artemi Panarin, Mika Zibanejad, Alexis Lafreniere and Kaapo Kakko, defenseman Adam Fox and goalie Igor Shesterkin.