Cohen was born in Boston, the son of a furrier from England, and was the first in his family with a college education. He graduated from Tufts University in 1951. He was guided towards science instead of medicine in a chance encounter with Isaac Asimov while working a summer job as a waiter, and completed his Ph.D. at the University of Southern California in 1955, helping to support his graduate studies by working as an extra in the movies of Katharine Hepburn and Spencer Tracy. His doctoral research, supervised by Jerome A. Berson, concerned the synthesis of alkaloids found in ipecac, and the chemical properties of pyridines.
After postdoctoral research as a Fulbright scholar at the University of Glasgow, working with Derek Barton, he joined the University of Pittsburgh chemistry faculty in 1956, and became one of the first professors at the university to bring in federal grant money for his research. He retired as a professor emeritus in 1999, but continued to do research in his laboratory, often working 80-hour weeks.
At the University of Pittsburgh, he was the doctoral advisor to over 40 students.