Chiasson played for the Des Moines Buccaneers of the USHL for the 2008–09 season. While in Des Moines he was coached by J. P. Parisé. That year he led the team in scoring and was selected for the 2009 USHL All-Star Game. He was drafted in the 2009 NHL Entry Draft by the Dallas Stars in the second round.
In 2009 Chiasson enrolled at Boston University, then the defending NCAA Division I Ice Hockey National Champions. He has remarked that he enjoyed playing for BU in part due to the close knit nature of the team. In his first season at BU he missed several games due to a concussion. During the 2010 off season Chiasson worked out with Patrice Bergeron in Quebec. He was successful in the 2010–11 season, and led his team in scoring. While not playing hockey, Chiasson enjoys golfing and attending Boston Red Sox games. His teammates nicknamed him "chaser".
Professional
Dallas Stars
On March 26, 2012, Chiasson signed a three-year contract with the Dallas Stars organization. He was assigned to their AHL affiliate, the Texas Stars, where he finished the rest of the season. During the lock-out-shortened 2012–13 season, Chiasson was recalled from Texas and made his NHL debut on April 3, 2013 against the Anaheim Ducks. In his second career game Chiasson scored his first NHL goal on April 5, 2013 against Viktor Fasth of the Anaheim Ducks. Playing in the first line along with Jamie Benn and Ray Whitney, Chiasson scored six goals in his first six games.
Ottawa Senators
Chiasson was involved in a blockbuster trade on July 1, 2014, being dealt to the Ottawa Senators along with Alex Guptill, Nick Paul, and a 2015 second round pick in exchange for star forward Jason Spezza, and Ludwig Karlsson. He scored a goal in his first game with the Senators on October 9, 2014. Following the 2014–15 NHL season Chiasson became a restricted free agent under the NHL Collective Bargaining Agreement. The Ottawa Senators made him a qualifying offer to retain his NHL rights and, on July 5, 2015, Chiasson filed for salary arbitration. Chiasson was seeking $2.4 million per year, while the Senators offered $1 million. The arbitrator awarded Chiasson a one-year contract of $1.2 million.
Calgary Flames
Following the 2015–16 season, Ottawa and Chiasson again clashed over a new contract, unable to agree to a dollar value. As such, the Senators traded Chiasson to the Calgary Flames in exchange for Patrick Sieloff on June 27, 2016. He was promptly signed as a restricted free agent to a one-year $800,000 contract with the Flames. He rebounded offensively with the Flames in the 2016–17 season, providing depth scoring with 12 goals and 24 points in 81 games. As an arbitration eligible restricted free agent following a first-round exit in the post-season, Chiasson was not extended a qualifying offer by the deadline on June 26, 2017.
Washington Capitals
On September 9, 2017, the Washington Capitals signed Chiasson, then an unrestricted free agent, to a professional tryout. Following training camp and preseason, on October 4, 2017, the Capitals signed Chiasson to a $660,000 one-year contract. Chiasson won his first Stanley Cup in 2018 with the Capitals.
Edmonton Oilers
On September 10, 2018, Chiasson signed a professional try out contract with the Edmonton Oilers, and on October 2, signed a one-year contract with the Oilers. With the Oilers, Chiasson set a new career high for most goals in a season, scoring his 14th on December 14, 2018, against Anthony Stolarz of the Philadelphia Flyers in a 4-1 victory.
