Branscomb began his career as an instructor in the department of philosophy at Southern Methodist University. He became a professor of New Testament literature at Duke University in 1925. He was awarded a Guggenheim Fellowship in 1930-1931 to study in Berlin and Marburg, Germany. He was the director of the Duke University Libraries from 1933 to 1941, and later the dean of the Duke Divinity School. Branscomb served as the fourth chancellor of Vanderbilt University from 1946 to 1963. During his tenure, at least 18 new buildings were erected on campus, and the endowment went from $38 million to $88 million. In 1960, at the insistence of trustees, especially James Geddes Stahlman, Branscomb expelled James Lawson, an African-American student and Congress of Racial Equality leader who organized sit-ins in defiance of Nashville's segregation laws, from the Vanderbilt Divinity School. A dozen faculty members resigned in protest. Branscomb remained unapologetic about his decision as late as 1980. Branscomb served as first chairman of the United States Advisory Commission for Education Exchange from 1947 to 1951. From 1955 to 1958 he served as Commission on Education and International Affairs of the American Council of Education. From 1963 to 1964 he was an educational consultant for the World Bank, and he chaired the United States Commission for UNESCO from 1963 to 1965. He also served as vice-chairman of the United States Delegation to the Unesco General Conference in Paris in 1964, chair of the United States Delegation to the World Conference on the Eradication of Illiteracy in Tehran in 1965. He traveled to Geneva for the United States Delegation to the World Health Organization Assembly in 1965 and 1966, and to Buenos Aires to chair the U.S. Delegation to the Conference of Ministers of Education and Ministers in Charge of Planning in 1966. Branscomb was the author of several books about New Testament theology. He was the president of the National Association of Biblical Instructor in 1940. He served on the commission on church and war of the Federal Council of Churches, and on the American theological committee of the World Council of Churches. Branscomb served on the board of directors of the Association of Rhodes Scholars, and he was the editor of the American Oxonian in the 1940s. He was also on the board of the American Council of Learned Societies. Branscomb served on the editorial board of the South Atlantic Quarterly. He held honorary degrees from Brandeis University, Northwestern University, Southwestern University, and Hebrew Union College.
Branscomb married Margaret Vaughan, the daughter of a lawyer from Greenville, Texas and the niece of a Vanderbilt alumna, in 1921. They had three sons: Harvie, Ben and Lewis. They resided in Nashville, Tennessee. Branscomb played tennis and golf, and he raised gladioli and chrysanthemums. The Branscomb quad, on Vanderbilt's campus is named after him. Branscomb died on July 23, 1998 in Nashville. His funeral was held in Benton Chapel on the Vanderbilt campus, where he was buried in a vault with his wife.
