Elsasser was born in the German city of Mannheim and educated at the University of Göttingen, where he obtained his doctorate in 1927. He worked at the University of Frankfurt before leaving Germany in 1933 following Hitler's rise to power. He taught at the Sorbonne, Paris, before emigrating to America (1936) where he joined the staff of the California Institute of Technology. He became professor of physics at the University of Pennsylvania (1947–50) and at the University of Utah (1950–58). In 1962 he became professor of geophysics at Princeton and he was appointed research professor at the University of Maryland from 1968 until his retirement in 1974.

Elsasser made fundamental proposals on the question of the origin of the Earth's magnetic field. It had been known for some time that this could not be due to the Earth's iron core for its temperature is too high for it to serve as a simple magnet. Instead he proposed that the molten liquid core contains eddies set up by the Earth's rotation. These eddies produce an electric current that causes the familiar terrestrial magnetic field.

Elsasser also made predictions of electron diffraction (1925) and neutron diffraction (1936). His works include The Physical Foundation of Biology (1958) and Atom and Organism (1966).