Chiasson, a native of Montreal, began his road to the NHL when he left Quebec as a high school senior to attend Northwood High in Lake Placid, New York, and play hockey in the 2007-08 season.

He played 2008-09 with Des Moines of the United States Hockey League, where he was coached by former NHL forward J.P. Parise, and had 50 points in 56 games. That convinced the Dallas Stars to select Chiasson in the second round (No. 38) of the 2009 NHL Draft.

Chiasson enrolled at Boston University, where he played three seasons. After leading BU in scoring with 46 points (15 goals, 31 assists) in 2011-12, Chiasson signed with Dallas and was assigned to Texas of the American Hockey League, where he played nine games that season.

He started the 2012-13 season with Texas and played 57 games before being called up to the Stars. He made a quick impression with six goals and seven points in seven games.

Chiasson made the Stars at training camp in 2013-14 and finished with 35 points (13 goals, 22 assists) in 79 games, but was traded to the Ottawa Senators on July 1, 2014 as part of the package that sent center Jason Spezza to Dallas.

He played the next two seasons as a bottom-six forward before being traded to the Calgary Flames on June 27, 2016. After finishing with 24 points (12 goals, 12 assists) in 81 games with Calgary in 2016-17, Chiasson moved on to the Washington Capitals for the 2017-18 season, scoring nine goals and 18 points in 61 games. He also had a goal and an assist in 16 Stanley Cup Playoff games to help Washington win the Stanley Cup for the first time since entering the NHL in 1974.

Chiasson signed a one-year contract with the Edmonton Oilers on Oct. 2, 2018 and scored an NHL career-high 22 goals that season. He subsequently played for the Vancouver Canucks and Detroit Red Wings before retiring from the NHL on Jan. 6, 2025, after 11 seasons.