Swan was born in Fitchburg, Massachusetts to Finnish parents who had immigrated to the United States at the turn of the century; he was the second of nine children.

His father was a keen amateur musician and before Einar Swan had entered his teens, he played violin, clarinet, saxophone and piano.

[1] Around 1924, the bandleader Sam Lanin invited Swan to join his orchestra at New York's famed Roseland Ballroom, and Swan played with leading musicians such as cornettist Red Nichols, and members of The Charleston Chasers Vic Berton (drums) and Joe Tarto (tuba), with whom he soon started composing and arranging material for the orchestra.

Shortly thereafter, the Bar Harbor Society Orchestra released "Trail of Dreams" credited to Swan and Raymond Klages.

The song became a hit and has since been covered by many other performers such as Lee Wiley, Louis Armstrong, Ray Charles, Ethel Waters, Billie Holiday, Sarah Vaughan and Frank Sinatra.