Evins was a native of DeKalb County, Tennessee. He graduated from Vanderbilt University in Nashville, Tennessee in 1933 and the Cumberland School of Law in Lebanon, Tennessee in 1934. He was admitted to the bar in that same year and began practice in Smithville, Tennessee, county seat of DeKalb County.

In 1935 Evins was named a staff attorney for the Federal Trade Commission, and served in this position until 1938, when he was named the FTC's assistant secretary, a position which he held until 1940. Shortly after U.S. entry into World War II, he was commissioned in the United States Army Judge Advocate General Corps, serving on active duty until 1946, when he resumed his law practice in Smithville. Upon his return, he was also elected chairman of the Smith County Democratic Party. Later in that same year, he won the nomination of the Democratic Party for the seat from the 5th District. He won the election easily in this solidly-Democratic area, and was re-elected to fourteen more terms, generally with little or no opposition. His district was renumbered the 4th after the 1950 Census, when Tennessee lost a congressional district.

Evins was a powerful figure in Congress. He was chairman of the House Select Committee on Small Business for six years, and for the following Congressional session of the United States House Committee on Small Business, and served on the important House Appropriations Committee. He used his influence to make sure that his district, a mostly rural area east and south of Nashville, was well taken-care of; Smithville was the smallest city chosen for participation in the Model Cities Program and its major throughfare was renamed "Congressional Boulevard." Like most of his constituents, Evins was a moderately conservative Democrat; he was slow to accept to racial desegregation, not because of deep-seated personal bigotry but because it was a change to what had long been the accepted order of things. However, he was one of only two Tennessee Democratic congressmen not to sign the Southern Manifesto.

Evins decided to retire in 1976 after serving a total of 15 terms; in a spirited primary to succeed him, Al Gore won and began his political career.

Evins' family was and is very prominent in the area; one brother ran a local bank, another is the namesake of an area state park, another was the guiding force behind the founding of the Cracker Barrel restaurant chain. A niece, Karlen Evins, is a prominent Nashville radio personality.

Evins died in Nashville and is buried in the Smithville Town Cemetery in Smithville.