He attended public and private schools. Broussard attended the Catholic Georgetown University in Washington, District of Columbia, from 1879 to 1882. He graduated from the Tulane University Law School in 1889.
He was a night inspector of customs in New Orleans from 1885 to 1888, when he was appointed assistant weigher and statistician. He held that position in 1888-1889. He was admitted to the bar the same year and launched his practice in New Iberia.
He was elected prosecuting attorney of the Nineteenth Judicial District and held that office from 1892 to 1897.
Broussard was elected as a Democrat to the Fifty-fifth and to the eight succeeding Congresses (March 4, 1897 – March 4, 1915). While in the House of Representative, he was chairman of the Committee on Expenditures in the Department of Justice (Sixty-third Congress).
He did not seek renomination in 1914, having become a candidate for Senator. He was elected in 1914 to the Senate and served from March 4, 1915, until his death three years later in New Iberia.
While in the Senate he was chairman of the Committee on National Banks (Sixty-fourth and Sixty-fifth Congresses).
Interment was in the Catholic Cemetery in New Iberia. Edwin Sidney Broussard, I, Robert"s younger brother, also practiced law in New Iberia and was elected to succeed Robert in the United States. Senate. One of Broussard"s nephews, George P. Broussard (1916–1977), was a veterinary medical researcher in New Iberia.
Near the end of his Senate tenure, Broussard"s secretary was West. Burch Lee, the former state representative for Webster Parish and later clerk of the United States District Court for the Western District of Louisiana, based in Shreveport.