Asham was drafted 71st overall by the Montreal Canadiens in the 1996 NHL Entry Draft and made his NHL debut with Montreal during the 1998–99 season. After parts of four seasons with Montreal, he was traded to the New York Islanders on June 22, 2002, along with a 2002 5th round draft pick for Mariusz Czerkawski. After years of going back and forth between the Canadiens and their American Hockey League affiliate, Asham became a mainstay in the Islanders line-up with his most productive offensive season to date coming during his first season with his new team in 2002–03, when he scored 15 goals and 19 assists for 34 points in 78 games. After four seasons with the Islanders, Asham signed a one-year contract with the New Jersey Devils on August 7, 2007. After the 2007–08 season, Asham became a free agent again, this time signing a two-year contract with the Philadelphia Flyers on July 7, 2008. On signing with the Flyers, Asham said, "I've always loved the way the Flyers play and I fit in here. They play hard every night. They play tough hockey and I've always admired that. I always thought I should play here and when the chance came up last summer to come here, I thought it was the best place for me and a place where I'd really have a chance to win a Stanley Cup in the next couple of years." In the 2010 playoffs, Asham scored four goals as the Flyers defied expectations and reached the Stanley Cup Finalsfor the first time since 1997. His biggest goal came in Game 5 of the Eastern Conference Finals vs. Montreal when he scored in the 2nd period to give the Flyers a 2–1 lead of an eventual 4–2 victory that eliminated the Canadiens and propelled the Flyers into a Finals matchup against the Chicago Blackhawks. Asham netted one goal in the finals as the Flyers lost to the Blackhawks 4 games to 2. Asham signed a one-year contract worth $700,000 with the Pittsburgh Penguins on August 20, 2010. During the following off-season, the contract was extended for another year. During Game 3 of the 2012 NHL Eastern Conference quarterfinals, in retaliation to a hit on a teammate, Asham cross-checked Brayden Schenn of the Philadelphia Flyers in the upper body, and then threw a punch while Schenn was down on the ice. Schenn sustained no apparent injury due to the hit. For his actions, Asham received a four-game suspension, his first disciplinary action from the NHL Commissioner since joining the NHL. Asham signed a two-year contract with the New York Rangers on July 1, 2012. With this, Asham had played for all 5 teams in the former Atlantic division during his career. On February 10, 2013, Asham scored his first goal as a Ranger in a 5-1 victory over the Tampa Bay Lightning. He is also the only player to score a goal with every Atlantic division team. On May 13, 2013, he scored the series-winning goal in Game 7 of the 2013 Stanley Cup playoffs to push the Rangers into the second round and eliminate the Washington Capitals. Asham saw limited action with the Rangers during the 2013–14 season, playing in only six games. He spent most of the season with the Hartford Wolf Pack. The Rangers did not offer to extend Asham's contract and on July 1, 2014, he became an unrestricted free agent, effectively ending his professional career. In September 2015, Asham signed an amateur playing contract with the Gander Flyers of the Central West Senior Hockey League.
Personal life
Asham's hometown is Portage La Prairie, where he still has many friends and family. He has 4 children: daughters Azilyn and Oceane, and sons Dexter and Cruz. Asham is Métis Canadian. Asham also has his own charity called "Arron's Chance to Play Hockey" This foundation's mission is to work in partnership with the community to provide opportunities that will enhance the emotional, physical, social and intellectual well being of children. The overall goal is to enhance the lives of children and families.