Kantorovich was born on 19 January 1912, to a Russian Jewish family. His father was a Doctor practicing in Saint Petersburg. In 1926, at the age of fourteen, he began his studies at the Leningrad University. He graduated from the Faculty of Mathematics in 1930, and began his graduate studies. In 1934, at the age of 22 years, he became a full professor.
After 1939, he became the professor of Military engineering-technical university. During the Siege of Leningrad, Kantorovich was the professor of VITU of Navy and in charge of safety on the Road of Life. He calculated the optimal distance between cars on ice, depending on thickness of ice and temperature of the air. In December 1941 and January 1942, Kantorovich personally walked between cars driving on the ice of Lake Ladoga, on the Road of Life, to ensure the cars did not sink. However, many cars with food for survivors of the siege were destroyed by the German air-bombings. In 1948 Kantorovich was assigned to the atomic project of the USSR.
Since 1960, Kantorovich lived and worked in Novosibirsk, where he created and took charge of the Department of Computational Mathematics in Novosibirsk State University.