Born in San Marcos, Texas, in 1863, Burleson was raised there in a military family. His father was Confederate officer during the Civil War. Burleson attended Texas A&M and Baylor University, earning a law degree from the University of Texas at Austin in 1884. After a brief stint practicing law, Burleson was elected as both the Austin assistant city attorney and the district attorney for the Twenty-sixth Texas Judicial District.

A wealthy man, Burleson won an election to the U.S. House of Representatives as a Democrat and served from 1899 to 1913. He was on the House Agriculture, Census, and Foreign Affairs Committees and was a ranking member of the House Appropriations Committee. Burleson advised the 1912 Democratic presidential candidate Woodrow Wilson as part of Wilson's "veranda cabinet."Although wary of Burleson's conservatism, President Wilson appointed Burleson postmaster general, the first native Texan ever to serve in the cabinet. While postmaster general, Burleson segregated African American and white workers and downgraded or fired southern blacks in the postal service; at the same time, he increased rural mail service and initiated airmail service. A staunch proponent of the Espionage Act during World War I, Burleson used his power to suppress certain kinds of expression. Following his time in government, Burleson returned to pursue agriculture in Austin, dying there in November 1937 at the age of 74.