Brennan was born in Mount Clemens and moved with his parents to Detroit in 1895. He graduated from SS. Peter and Paul’s Parochial School, then from Detroit College in 1909, from the law department of Harvard University in 1912, and from the University of Detroit in 1914. He was admitted to the bar in 1912 and commenced practice in Detroit. He was legal adviser to the Michigan State Labor Department in 1912 and 1913; assistant corporation counsel for the city of Detroit 1915-1920; member of the Michigan State Senate from the 2nd District in 1919 and 1920. He drafted the automobile traffic ordinance of Detroit, used as a model for many other cities.

Brennan was elected as a Republican to the United States House of Representatives from Michigan's 13th District for the Sixty-seventh Congress, serving from March 4, 1921 to March 3, 1923. He was not a candidate for reelection in 1922. He was elected judge of the circuit court of Wayne County, for the term commencing in January 1924 and was reelected for six successive terms, serving served until his resignation effective December 31, 1954. He then practiced law until his death in Detroit. He is interred in Holy Sepulchre Cemetery in Birmingham.

Brennan introduced a bill in 1922, the first which proposed to allow radio coverage of U.S. House of Representatives proceedings. The bill failed, and the idea was not revived until the 1940s.