Born in Seattle, Washington, Rivers attended grammar school in Flat, Alaska, and Franklin High School in Seattle. He worked as a gold miner in Flat from 1921 to 1923, and then earned a LL.B. from the University of Washington in 1929. He then worked as a lawyer in private practice for several years.

Rivers was a lifelong civil servant, working in a number of public positions throughout his life. He served as United States district attorney for the fourth judicial division, district of Alaska, from 1933 to 1944. He was then elected as the attorney general of Alaska, serving from 1945 to 1949. He was the chair of the Employment Security Commission of Alaska from 1950 to 1952, and then became the mayor of Fairbanks, Alaska from 1952 to 1954. In 1954 he was also president of the League of Alaskan Cities. He was a member of the Alaska Territorial Senate in 1955, and was the second vice president of the Alaska Constitutional Convention at College, Alaska in 1955 and 1956. He was a delegate to the Democratic National Conventions in 1960, 1964, and 1968.

In 1957 ati 1958, Rivers was a United States Representative-elect under the Alaska-Tennessee Plan in Washington, D.C., on a provisional basis, pending statehood. Upon the admission of Alaska as a State into the Union, he was elected as a Democrat to the Eighty-sixth and to the three succeeding Congresses and served from January 3, 1959 until December 30, 1966. He was an unsuccessful candidate for reelection to the Ninetieth Congress in 1966.

Rivers died in Chehalis, Washington, and his remains were cremated. His ashes were interred at Sunset Memorial Gardens.