Eigen, the son of a musician, was born at Bochum in Germany and educated at the University of Göttingen where he obtained his PhD in 1951. He joined the staff of the Max Planck Institute for Physical Chemistry at Göttingen in 1953 and served as its director from 1964.

In 1954 Eigen introduced the so-called relaxation techniques for the study of extremely fast chemical reactions (those taking less than a millisecond). Eigen's general method was to take a solution in equilibrium for a given temperature and pressure. If a short disturbance was applied to the solution the equilibrium would be very briefly destroyed and a new equilibrium quickly reached. Eigen studied exactly what happened in this very short time by means of absorption spectroscopy. He applied disturbances to the equilibrium by a variety of methods, such as pulses of electric current, sudden changes in temperature or pressure, or changes in electric field.

The first reaction he investigated was the apparently simple formation of a water molecule from the hydrogen ion, H⁺, and the hydroxide ion, OH^–. Calculations of reaction rates made it clear that they could not be produced by the collision of the simple ions H⁺ and OH^–. Eigen went on to show that the reacting ions are the unexpectedly large H₉O₄ ⁺ and H₇O₄ ^–, a proton hydrated with four water molecules and a hydroxyl ion with three water molecules. For this work Eigen shared the 1967 Nobel Prize for chemistry with George Porter and Ronald Norrish.

Eigen later applied his relaxation techniques to complex biochemical reactions. He also become interested in the origin of nucleic acids and proteins; with his colleague R. Winkler he proposed a possible mechanism to explain their formation. Much of this and subsequent work was described by Eigen in his Laws of the Game: How Principles of Nature Govern Chance (1982).