Einstein was born in Munich. Though he originally studied law, he quickly realized his principal love was music, and he acquired a doctorate at Munich University, focusing on instrumental music of the late Renaissance and early Baroque eras, in particular music for the viola da gamba. In 1918 he became the first editor of the Zeitschrift für Musikwissenschaft; slightly later he became music critic for the Münchner Post; and in 1927 became music critic for the Berliner Tageblatt. In this period he was also a friend of the composer Heinrich Kaspar Schmid in Munich and Augsburg. In 1933, after Hitler's rise to power, he left Nazi Germany, moving first to London, then to Italy, and finally to the United States in 1939, where he held a succession of teaching posts at universities including Smith College, Columbia University, Princeton University, the University of Michigan, and the Hartt School of Music in Hartford, Connecticut.

Einstein not only researched and wrote detailed works on specific topics, but wrote popular histories of music, including the Short History of Music (1917), and Greatness in Music (1941). In addition, he published a revision of the Köchel catalog of Mozart's music (1937), and a comprehensive, three-volume set The Italian Madrigal (1949) on the secular Italian form, the first detailed study of the subject. His 1945 volume Mozart: His Character, His Work was an influential study of Mozart and is perhaps his best known book.

From circa 1922 he influenced Shinichi Suzuki (an amateur violinist visiting Berlin from Japan, and inventor of the Suzuki method of early learning).

While one respected source lists Alfred as a cousin of the scientist Albert Einstein, another claims that no relationship has been verified. Some Web sites claim they were both descended from a Moyses Einstein seven generations back, hence they were sixth cousins. Einstein's daughter Eva maintains that they were not related.