A native New Yorker, Cohen was educated at Brooklyn and Oberlin colleges and at the University of Michigan, where he was appointed teaching fellow in the department of biochemistry in 1946. He moved to the University of Colorado School of Medicine in 1948, and in 1952 he took up the post of American Cancer Society postdoctoral fellow at Washington University, St. Louis. His long association with the Vanderbilt University School of Medicine, Nashville, began in 1959, with his appointment as assistant professor of biochemistry. He subsequently became associate professor (1962), professor (1967), and distinguished professor (1986).

Cohen's appointment to Washington University in 1952 marked the start of a fruitful collaboration with the Italian cell biologist, Rita Levi-Montalcini, who had discovered a chemical produced by a culture of mouse tumor cells that influenced the number of nerve cells growing in chick embryos. Cohen set about trying to characterize this growth factor (later termed nerve growth factor), and found the same chemical in snake venom and in the salivary glands of adult male mice.

His findings led Cohen to investigate another growth factor that influences the embryological development of such tissues as those of eyes and teeth, which are derived from epidermis. He was able to identify a receptor on the cell membrane that was responsive to this epidermal growth factor. This was of great significance, suggesting a mechanism by which cells are able to interact with chemical messengers such as hormones, which control their growth or normal functions. Such cell-surface receptors are also a crucial element in the abnormal uncontrolled growth of cells in cancer.

For his work on growth factors and membrane receptors, Cohen was awarded the 1986 Nobel Prize for physiology or medicine, jointly with Levi-Montalcini.