Norris Cotton was born on a farm in Warren, New Hampshire. He was educated at Philips Exeter Academy in New Hampshire and Wesleyan University in Connecticut. While in college, he served as a clerk to the New Hampshire state senate and as a member of the New Hampshire state Assembaly in 1923 as one of the youngest legislators in history. He became a lawyer and practiced law in Lebanon, New Hampshire. He was elected to the state assembly again in 1943. He served as majority leader that year and as speaker during 1945.

In 1946 he was elected to the United States House of Representatives from New Hampshire for the first time. He served until 1954 when he ran for a seat in the United States Senate from New Hampshire in a special election to fill the vacancy caused by the death of Charles W. Tobey. He was elected to a full term in 1956, reelected twice and served in the Senate until 1975.

One of his most controversial decisions was his vote against the Civil Rights Act of 1964, which all other senators in Cotton's region of the country voted for. However, Cotton would vote for later civil rights acts like the Voting Rights Act of 1965 and the Civil Rights Act of 1968. He was a prominent leader of his party in the Senate, chairing the Senate Republican Conference from 1973 to 1975. He did not run for reelection in 1974.

He was reappointed to the Senate in August 1975 after the election of his successor was contested. As the closest Senate election in history, it was subjected to many recounts, but both candidates refused to give up. Cotton served as a temporary senator until a special election in September 1975, which was less close. Cotton returned to Lebanon, New Hampshire, where he died at age 88 from natural causes.

The comprehensive cancer center at Dartmouth-Hitchcock Medical Center in Lebanon is named for Senator Cotton.