Asbjornson batted and threw right-handed. He was born in Concord, Massachusetts. Asbjornson was 19 years old when he debuted with the Boston Red Sox in 1928, being one of seven catchers used by the Red Sox in an unsuccessful attempt to replace retired Grover Hartley.
Later, he saw more action with the Cincinnati Reds as a backup for Clyde Sukeforth and Ernie Lombardi in part of two seasons.
His best year was 1931, when he posted career-highs in batting average (305), runs batted in (22) and games played (45). In a four-season career, Asbjornson was a.235 hitter with one home run and 27 Reserve Bank of India in 97 games.
Asbjornson died in Williamsport, Pennsylvania, at the age of 60.