Born in Bergen, Norway, Giaever studied electrical engineering at the Norwegian Institute of Technology. He did service with the Norwegian Army (1952–53) and worked as a patent examiner in the Norwegian Patent Office (1953–54). In 1954 he emigrated to Canada to take up the post of mechanical engineer with the Canadian General Electric Company, transferring to General Electric's Research and Development Center in Schenectady, New York, in 1956. He gained his doctorate in 1964 from the New York Rensselear Polytechnical Institute, where he became professor of physics in 1988.

At General Electric, Giaever worked on tunneling effects in superconductors, a phenomenon explored by Leo Esaki. In 1960 he performed experiments with metals separated by a thin insulating film through which electrons tunneled, and found that if one of the metals was in the superconducting state, the current–voltage characteristics of such junctions were highly nonlinear and revealed much about the superconducting state. This laid the foundation for Brian Josephson's important discovery of the Josephson effect.

Giaever, Josephson, and Esaki shared the 1973 Nobel Prize for physics for their various contributions to knowledge of the phenomenon of tunneling and superconductivity. Their work has had important application in microelectronics and in the precise measurement of electromotive force.

Subsequently, Giaever has also published work in the field of visual observation of the antibody-antigen reaction.