Klein was born in Omaha, Nebraska. He received his bachelor's degree from the University of California, Berkeley, in 1942 and his PhD from the Massachusetts Institute of Technology in 1944. He became a research associate at the Cowles Commission at the University of Chicago (1944-47), then at the National Bureau of Economic Research in New York (1948-51) and finally at the Survey Research Center of the University of Michigan (1949-54), where he taught from 1950. In 1954 he left the USA in protest against the activities of the McCarthy Committee, for a readership at the Oxford Institute of Statistics. In 1958 he returned to the USA to take up a professorship at the University of Pennsylvania, and remained there until his retirement from teaching in 1990.

Klein was awarded the John Bates Clark Medal of the American Economic Association in 1959 and was president of the Econometric Society in 1960, the Eastern Economic Association in 1974, the Environmental Economics Association in 1975, the American Economic Association in 1977, and the Atlantic Economic Society in 1990. He served as chair of the economic task force of Democratic presidential candidate Governor Carter.

His works include The Keynesian Revolution (1947), Economic Fluctuations in the United States, 1921-1941 (1950), Econometric Model of the United States 1929-52 (1955; with A S Goldberger), Econometric Model of the United Kingdom (1961; with R J Ball and others), Textbook of Econometrics (1962), History of Macroeconomic Model-Building (1991; with R G Bodkin and K Marwah), and The Economics of Supply and Demand (1983).