Burrows played two seasons in the Quebec Major Junior Hockey League (QMJHL) with the Shawinigan Cataractes, beginning in 2000–01.
The following season, in 2003–04, he returned to the South Division, as he was signed by the Columbia Inferno.
Early in the season, he was signed by Columbia's American Hockey League (AHL) affiliate, the Manitoba Moose, on October 21, 2003, He appeared in two AHL games for Manitoba before being sent back down to the ECHL.
The following season, he improved to 35 goals and 70 points over 64 games, third in team-scoring, behind Jonathan Bellemare and Jason Pominville.
as the Cataractes advanced to the Conference Finals, where they were eliminated in seven games by the Victoriaville Tigres.
In 2007–08, Burrows formed an effective duo with centre Ryan Kesler on the third line as defensive forwards, countering opposing teams' top players while contributing offensively, as well.
After remaining on the third line with Kesler at the start of the following season, head coach Alain Vigneault separated the two after the All-Star break, placing Burrows on the first line with Daniel and Henrik Sedin, beginning on February 12, 2009, during a game against the Phoenix Coyotes.Burrows established himself as a checking forward with the Canucks in his first three NHL seasons before emerging as a scorer with four consecutive 25 goal seasons from 2008–09 to 2011–12.Before making it to the NHL, Burrows also enjoyed a prolific ball hockey career, competing in national and international tournaments in the summers.Burrows' ascension to the NHL has been attributed to his hard-working and abrasive style of play, generating momentum for his team and aggravating opposing players.He struggled to produce offensively, however, and recorded a career-low three goals and nine points in 81 games.Playing in a more offensive role on the first line for the latter part of the season, Burrows finished the 2008–09 campaign with 51 points.