Griffin grew up in Binghamton, New York, and played with the bands of saxophonist Charlie Barnet and Rudy Vallee and the CBS Radio Orchestra, before joining the clarinetist's famous ensemble in 1936. When Goodman played at Carnegie Hall in January 1938—in an event that is credited with transforming jazz into concert music— Harry James, Ziggy Elman, and Griffin made up the trumpet section.
Griffin left the Goodman band in 1939 to spend time with his family—he had married Helen O'Brien, a singer with the Tommy Dorsey Orchestra, in 1936—and returned to CBS. He spent the rest of his career as a studio musician, recording with Frank Sinatra, Billie Holiday, Charlie Parker, Louis Armstrong, and others. He also appeared on CBS television and radio, performing on the Ed Sullivan and Jackie Gleason shows.