Griffin was born in Detroit, Michigan and attended public schools in Garden City and Dearborn. During the Second World War, he enlisted in the U.S. 71st Infantry Division in 1943 and spent fourteen months in Europe. After the war, he graduated from Central Michigan College at Mount Pleasant in 1947. He received a law degree from the University of Michigan Law School and was admitted to the bar in 1950. He commenced the practice of law in Traverse City.

Griffin was elected as a Republican to U.S. House of Representatives from the 9th District for the Eighty-fifth and to the four succeeding Congresses, serving from January 3, 1957, until his resignation May 10, 1966. He was appointed on May 11, 1966, to the United States Senate to fill vacancy caused by the death of Patrick V. McNamara. He was elected November 8, 1966, to a full six-year term commencing January 3, 1967 and was reelected in 1972, serving from May 11, 1966, to January 2, 1979. He was Republican Whip 1969-1977. He was an unsuccessful candidate for reelection in 1978, defeated by Democrat Carl Levin. He was a justice of the Michigan Supreme Court 1987-1994. His son, Richard Allen Griffin, was a judge on the Michigan Court of Appeals, 1989-2005, until he was appointed by President George W. Bush to the United States Court of Appeals for the Sixth Circuit in 2005.

Robert Griffin is a resident of Traverse City.