Armstrong joined the Dallas Stars organization in 1991 and was appointed general manager on January 25, 2002. He won the Stanley Cup as an assistant general manager with the Stars in 1999. Armstrong replaced Bob Gainey as GM of the Stars with 32 games to go in the 2001–02 NHL season, and one of his first moves was hiring Dave Tippett as head coach. Armstrong is the son of NHL Hall of Fame linesman Neil Armstrong, inducted in 1991.

On November 13, 2007, in the wake of a 7–7–3 start and a colossal meltdown by the team away against the Los Angeles Kings (losing 6–5 in overtime after leading 4–0 with seven minutes remaining in the game), Armstrong was fired as GM and replaced by former Stars player Brett Hull and assistant GM Les Jackson as interim co-general managers. During Armstrong’s tenure, the Stars went 210–109–35–23 in the regular season, representing a .634 winning percentage, which is best in Stars history for a General Manager. Some important moves Armstrong made during his time as GM included the drafting of players like Trevor Daley, James Neal, Loui Eriksson and Jamie Benn. He also made a trade for Mike Ribeiro in exchange for defenceman Janne Niinimaa, which turned out to be beneficial for Dallas’ offence.

On May 29, 2008, the St. Louis Blues announced they had named Armstrong as the club’s director of player personnel after signing him to a two-year contract. When the two years were up, Armstrong became the club’s general manager after Larry Pleau’s retirement.

On June 17, 2010, the Blues acquired goaltender Jaroslav Halak from the Montreal Canadiens in exchange for prospects Lars Eller and Ian Schultz. Although he had yet to officially take over duties as GM, this was considered to be Armstrong’s first big move as the GM of the Blues.

Armstrong has been part of Hockey Canada’s management group for the gold medal-winning teams in the 2010 and 2014 Winter Olympics; 2007 and 2016 Gold Medal World Championship Teams, 2016 World Cup Championship Winning Team plus the Silver Medal Winning Teams in the 2008 and 2009 World Championship. After winning the Stanley Cup in 2019 with the St. Louis Blues Armstrong became the 1st person in management to become part of the Double Triple Gold Club ( Winning Stanley Cup in 1999 and 2019, Olympics in 2010 and 2014 plus World Championship in 2007 and 2016). He also served on Canada’s 2002 World Championship Team (did not medal).

In 2013, Armstrong was named the NHL General Manager of the Year and runner up again in 2019.

In Armstrong’s ten seasons as General Manager with St. Louis, the Blues have won three Central Division Regular Season titles (2011-12, 2014-15, 2019-20) and reached the Stanley Cup Playoffs eight times, Divisional Finals four times, Conference Finals twice and won the Stanley Cup in 2019.