Berg was born in Brooklyn, New York, the son of Sarah Brodsky, a homemaker, and Harry Berg, a clothing manufacturer. Berg graduated from Abraham Lincoln High School in 1943, received his Bachelor of Science degree in biochemistry from Penn State University in 1948 and Ph.D. in biochemistry from Case Western Reserve University in 1952. He is a member of the Beta Sigma Rho fraternity (now Beta Sigma Beta).

After completing his graduate studies, Berg spent two years (1952–1954) as a postdoctoral fellow with the American Cancer Society, working at the Institute of Cytophysiology in Copenhagen, Denmark and the Washington University School of Medicine, and spent additional time in 1954 as a Scholar in Cancer Research with the Department of Microbiology at the Washington University School of Medicine. He worked with Arthur Kornberg, while at Washington University. Berg was also tenured as a research fellow at Clare Hall, Cambridge. He was a professor at Washington University School of Medicine from 1955 until 1959. After 1959, Berg moved to Stanford University, where he taught biochemistry from 1959 until 2000 and served as Director of the Beckman Center for Molecular and Genetic Medicine from 1985 until 2000. In 2000 he retired from his administrative and teaching posts, continuing to be active in research.

He was elected a Fellow of the American Academy of Arts and Sciences in 1966. In 1983, Ronald Reagan presented Berg with the National Medal of Science. He was elected a Foreign Member of the Royal Society (ForMemRS) in 1992. In 2005 he was awarded the Biotechnology Heritage Award by the Biotechnology Industry Organization (BIO) and the Chemical Heritage Foundation. In 2006 he received Wonderfest's Carl Sagan Prize for Science Popularization.

Berg is a member of the Board of Sponsors of The Bulletin of the Atomic Scientists [1]. He was also an organizer of the Asilomar conference on recombinant DNA in 1975. The previous year, Berg and other Scientists had called for a voluntary moratorium on certain recombinant DNA research until they could evaluate the risks. That influential conference did evaluate the potential hazards and set guidelines for biotechnology research. It can be seen as an early application of the precautionary principle.

Berg was awarded one-half of the 1980 Nobel Prize in Chemistry, with the other half being shared by Walter Gilbert and Frederick Sanger. Berg was recognized for "his fundamental studies of the biochemistry of nucleic acids, with particular regard to recombinant DNA", while Sanger and Gilbert were honored for "their contributions concerning the determination of base sequences in nucleic acids."

Berg is currently a Professor Emeritus at Stanford. As of 2000, he stopped doing active research, to focus on other interests, including involvement in public policy for biomedical issues involving recombinant DNA and embryonic stem cells and publishing a book about Geneticist George Beadle.