Burrows was born in North East, Pennsylvania, and moved then with his parents to Ashtabula County, Ohio.

Burrows raised an infantry company in 1862 to fight in the American Civil War and served as its captain until the fall of 1863.

In an 1876 debate, Burrows, an admired orator, spoke forcefully in favor of a bill intended to protect the civil rights of freed black slaves.

Unaware of the full import of Burrows' speech, Elkins shook his colleague's hand in congratulations, a gesture that many southern congressmen interpreted as support for the civil rights legislation.

As a result, Elkin's handshake with Burrows is blamed for costing New Mexico several Southern Democratic votes which had been needed to achieve statehood.

Burrows was elected as a Republican to the United States Senate to fill the vacancy caused by the death of Francis B. Stockbridge and was reelected in 1899 and 1905, serving from January 24, 1895, to March 3, 1911.