Griffiths was ordained a priest in Rome on March 12, 1927, for the Diocese of Brooklyn by Archbishop Giuseppe Palica.[3] Following his return to Brooklyn, the diocese assigned Griffiths as curate at St. Joseph's Parish in Babylon, New York.[1] He then served at Our Lady of Mercy Parish in Queens (1928 to 1929) and at St. Augustine's Church in Brooklyn (1929 to 1931).[2] In addition to his pastoral duties, Griffiths worked as an instructor of New Testament scripture at the Dominican Normal School in Amityville, New York, from 1927 to 1929.[2]
Griffiths was named secretary of the diocesan tribunal in 1929, and then he was named its auditor in 1931.[1] From 1935 to 1943, he served as vice-chancellor of the diocese.[2] He also served as diocesan censor of books.[1] The Vatican elevated Griffiths to the rank of papal chamberlain in 1938, and domestic prelate in 1944.[2] One of the co-founders of the Canon Law Society of America in 1939, Griffiths served as its president from 1941 to 1942.[1]
Following the American entry into World War II in 1941, Bishop Thomas Molloy placed Griffiths in charge of supervising the military chaplains from the diocese.[1] He also directed the work of the Vatican Information Service in the diocese, transmitting messages monthly to and from persons in enemy-occupied countries.[1] In November 1943, the Vatican named Griffiths as chancellor of the Military Ordinariate, serving Catholics in the U.S. armed forces throughout the world.[2]