Sandrich was born in New York City, to a Jewish family. He was an engineering student at Columbia University when he began in the film business by accident. While visiting a friend on a film set, he saw that the director had a problem in setting up a shot; Sandrich offered his advice. It worked. He then entered into the movies in the prop department, and became a director specializing in several comedy shorts in 1927. He then made his first feature the next year, but returned to shorts after sound arrival. In 1933, he directed the Academy Award-winning short, So This Is Harris!. He later returned to feature films, most notably comedies, starring the team of Bert Wheeler and Robert Woolsey in Hips, Hips, Hooray!. In 1934, Sandrich soon gained his first directorial assignment with the Fred Astaire-Ginger Rogers musical The Gay Divorcee, which proved a success.
The following year, he directed Top Hat, which is often thought the best of the Astaire/Rogers musicals. He continued working with the team on Follow the Fleet (1936), Shall We Dance (1937), and Carefree (1938). In 1940, Sandrich left RKO for Paramount, which offered him a chance to be not only a director but as well as a producer. He made other several successful films in this capacity, including two with Jack Benny, Buck Benny Rides Again and Love Thy Neighbor (both 1940), and the romantic comedy Skylark (1941), starring Claudette Colbert and Ray Milland. However, while all these were hits, it is Holiday Inn in 1942 starring Fred Astaire and Bing Crosby, with music by Irving Berlin, that is most remembered today, for it introduced the song "White Christmas" performed by the crooner Crosby which remained the biggest selling popular song for fifty-two years. So Proudly We Hail! was a Sandrich-produced and directed adaptation of the hit play. It was extremely popular and successful, and featured a pair of performers – Adrian Booth and George Reeves – whom Sandrich had intended to bring to stardom after the war.
In 1945, while in pre-production on a follow-up to Holiday Inn called Blue Skies, starring Bing Crosby and featuring Irving Berlin's music, and serving as president of the Directors Guild, Sandrich died suddenly, of heart failure. He was at that time one of the most trusted and influential directors in Hollywood, respected by his colleagues and the studio management. His interment was at Home of Peace Cemetery.