A veteran of the big band era going back to the 1940s, Alexander played with Jimmy Dorsey, Artie Shaw, and Benny Goodman. As a studio musician, he backed Tony Bennett, Perry Como, Teresa Brewer, Frank Sinatra, Eartha Kitt, Sarah Vaughan, Aretha Franklin, and Lena Horne. He played behind the The Ray Charles Singers on their 1968 At the Movies album of film songs, and with Bob Haggart on his 1968 album Big Noise from Winnetka.