Following two solid seasons in the BCJHL with Penticton, Kobasew spent one year at Boston College where he was a Hockey East Second Team All-Star, the Hockey East Rookie of the Year, a member of the NCAA Championship All-Tournament Team and the NCAA Championship Tournament MVP while leading Boston College to the 2001 NCAA Championship.

Even with all the success he had with Boston College in his Freshman year, Kobasew felt he would be better off joining the WHL's Kelowna Rockets in 2001-02. The Osoyoos, British Columbia native spent one season with Kelowna before making his professional debut in 2002-03. In his rookie season with the Flames, Kobasew played in 23 games while seeing most of his action with the team's AHL affiliate in Saint John where he registered 33 points (21-12-33) in 48 games before becoming earning a regular roster spot in 2003-04.

Kobasew became a regular in the Flames line-up in the 2005-06 NHL regular season and produced 31 points. However, the following season Kobasew would only tally 4 goals in his first 40 games. As a result, the Flames dealt the right winger with Andrew Ference to the Boston Bruins in exchange for Brad Stuart and Wayne Primeau in February of 2007.

Though his time is Boston was successful and productive, it would also prove to be short-lived. Early in the 2009-10 season, Kobasew was shipped to the Minnesota Wild in exchange for defenseman Craig Weller and a draft pick.

After two seasons in Minnesota in which he scored nine goals in each, Kobasew was once again a free agent in the summer of 2011. On July 1 he inked a deal with the Colorado Avalanche.