Bergström was born in Stockholm and educated at the Karolinska Institute there, where he obtained his MD in 1943. In 1947 he was appointed to the chair of biochemistry at Lund. In 1958 he moved to a comparable position at the Karolinska Institute, which he left in 1981.

In the 1930s Ulf von Euler found an active substance in human semen capable of lowering blood pressure and causing muscle tissue to contract. He named it prostaglandin on the assumption that it came from the prostate gland. It soon became clear that there was not one such substance but a good many closely related ones with a variety of important physiological roles, but as they were produced in small quantities and rapidly broken down by enzymatic action, they proved to be very difficult to isolate and analyze. From 100 kilograms of rams' seminal vesicles, Bergström was able to extract a minute dose. To his surprise, however, he found the prostaglandin “extraordinarily active in virtually non-existent doses.”

In the 1950s Bergström succeeded in extracting the prostaglandins referred to as PGD₂, PGE₂, and PGF₂. He went on to demonstrate that they were derived from arachidonic acid (C_{2 0}H_{3 6}O₂), a fatty acid present in the adrenal gland, liver, and brain. Bergström's discovery opened up the study of prostaglandins by allowing them to be produced in the laboratory. For his pioneering work in this field he shared the 1982 Nobel Prize for physiology or medicine with John Vane and Bengt Samuelsson.