Beckenbach was born July 18, 1906, in Oak Cliff, Dallas County, Texas, the son of a leather worker and on his father's side the grandson of immigrants from Germany.

In 1924, he began study at Rice University, where in 1929 he earned a master's degree and in 1931 a PhD under the direction of Lester R. Ford.

Beckenbach was also a leader in the founding (in 1948) of the Institute of Numerical Analysis, which was then a branch of the National Bureau of Standards.

From 1949 to 1963, he was a consultant for the Rand Corporation and in the academic year 1951/1952 he was a visiting professor at the Institute for Advanced Study.

He was a co-author of several college textbooks in mathematics, including algebra, trigonometry, and analytic geometry.