Angell graduated in 1849 from Brown University, Providence, R.I., and was professor of modern languages and literature there from 1853 to 1860. He served as president of the University of Vermont, Burlington, from 1866 to 1871. He assumed the presidency of the University of Michigan, Ann Arbor, in 1871. In his years of service, which ended in 1909, he established the first system of admission requirements for medical schools (1874), the first American chair in the science and art of teaching (1879), and the first instruction in forestry (1882). Angell served as U.S. minister to China (1880–81) and Turkey (1897–98), and he was also a member of the Canadian-American Deep Waterways Commission (1896–97).

In addition to his work at the University of Michigan, his achievements in active diplomacy and his scholarship in international law made him one of the most prominent and highly respected educators of his time. Besides contributions to periodicals, he wrote Progress in International Law (1875), Reminiscences (1912), and Selected Addresses (1912).