Glasser was educated at University of Southern California through an Alchin Chair Foundation scholarship. In 1937 he won the California Composers Contest and in 1945 he won the Southern California Contest. Albert Glasser began his career as a copyist in the music department at Warner Brothers in the late 1930s, learning the art of film scoring while working under Max Steiner and Erich Wolfgang Korngold. He later worked on orchestration, and by the mid-1940s was composing and directing his own scores. A hard and fast worker, Glasser found his musical skills put to the test in the frantic, down-to-the-wire world of B-picture making. He scored 135 movies between 1944 and 1962, not counting at least 35 features for which he received no credit.
In addition to scoring 300 television shows and 450 radio programs, he arranged and conducted for the noted American operetta composer Rudolf Friml and orchestrated for Ferde Grofe (with whom he first collaborated on the sci-fi classic Rocketship X-M). For the US War Department, Glasser composed for Frank Capra's Special Services Unit and for Office of War Information radio shows for overseas broadcasts. For television, he composed the score for the early and much admired western series, The Cisco Kid. For radio, he composed scores for Hopalong Cassidy, Clyde Beatty, and Tarzan. Glasser joined ASCAP in 1950, and his popular song compositions include "Urubu", "The Cisco Kid", "Someday" and "I Remember Your Love". In addition to his composition work, Glasser was an amateur radio operator (K6RFU).