Bergman, who was born in Katrineberg, Sweden, studied at the University of Uppsala, at first reading law and theology before turning to science and mathematics. He was a prolific scientist, working in physics, mathematics, and physical geography as well as chemistry. After graduating with a master's degree in 1758, he became professor of mathematics at Uppsala in 1761 and later professor of chemistry and pharmacy in 1767.

Bergman carried out many quantitative analyses, especially of minerals, and he extended the chemical classification of minerals devised by Axel Cronstedt. He remained an adherent of the phlogiston theory and although he firmly supported the doctrine of constant composition his analyses were not as solidly based as those of his later compatriot Jöns Berzelius. His most influential work was probably Disquisitio de Attractionibus Electivis (1785; A Dissertation on Elective Attractions). He compiled extensive tables listing relative chemical affinities of acids and bases. Bergman gave early encouragement to Karl Scheele, some of whose work he published.