Elihu S. Williams was born near New Carlisle, Ohio, the son of Henry Williams and Elizabeth Pettigrew, both of Virginia. Elihu Williams worked on his father's farm until 16 years of age, receiving education in the winter at public schools. Not satisfied, he demanded of his father to be sent regularly to school. His father told him if he wanted a better education than he was getting at home, to get it himself. With $1.50 in his pocket, Elihu Williams started out in life for himself.

He worked among the farmers to earn money to pay his board for a few months, then attended Antioch College to pass examination for a teacher's certificate. He taught school the following winter in Brandt, Ohio. In 1858, he commenced reading law in the office of F. P. Cuppy, Esq., of Dayton, Ohio, continuing to teach in winters until February 1861, when he was admitted to practice of law by the Supreme Court of Ohio.

In October 1861, he enlisted as a private in the 71st Regiment, Ohio Volunteer Infantry serving for the duration of the Civil War. He was commissioned first lieutenant February 14, 1862. He was in the Battle of Shiloh. He was promoted to captain February 10, 1863. In fall of 1863, he was detailed to the command of the military post at Carthage, Tennessee and remained there until the close of the war.

After the war, Capt. Williams remained in Smith County, Tennessee, and engaged in the practice of law. He took an active part in the first convention for the reconstruction of Tennessee at Nashville. In April 1865, he was commissioned District Attorney for the Sixth Judicial District of Tennessee until the summer of 1867. He resigned to accept the nomination as Republican candidate for the Tennessee House of Representatives, to represent the district of Sumner, Smith and Macon Counties, serving for two years in the Radical Legislature of Tennessee until 1869. He returned to agricultural pursuits at his farm.

On May 31, 1866 Elihu Williams married Alice Gordon, daughter of Dr. Wiley B Gordon and Virginia Russwum. They had 2 children; a third child enumerated as son in 1880 may have been adopted.

In 1875, he moved his family to Troy, Ohio, where he established partnership with his brother Henry H. Williams in the practice of law. He also became editor of the newspaper, the Buckeye. In 1886 he was elected as a Republican to the Fiftieth and Fifty-first Congresses from Ohio’s third district.

At the end of his congressional terms, he resumed the practice of law at Troy where he died. He is interred in Riverside Cemetery.