Curtis was born on his family's farm in Kearney County. He attended public schools and later attended Nebraska Wesleyan University, where he was a member of Theta Chi. He studied law on his own and passed the bar exam. He began practicing law in Kearney County and served as the country attorney from 1931 to 1934.

Curtis was elected to the House of Representatives in 1938 on an anti-Franklin D. Roosevelt and New Deal platform. He served in the House from 1939 until 1954, being reelected every two years. Curtis ran for the Senate from Nebraska in 1954 and won the election; he was reelected three more times to six-year times, serving from 1955 to 1979. He voted for the Civil Rights Act of 1964. Curtis was loyal to the Republican Party, particularly supporting its anti-communist stances and fiscal conservatism, which included opposition to social programs such the New Deal and Lyndon B. Johnson's Great Society.

Curtis was a close ally of both Barry Goldwater and Richard Nixon. He served as floor leader during the 1964 Republican National Convention, when Goldwater won the nomination.

He supported Nixon's Vietnam War escalation policy and remained loyal to him even during the height of the Watergate scandals, when Nixon was forced to resign as President. Curtis served as chairman of the Senate Republican Conference from 1975 to 1979.

Following his retirement, Curtis moved to Lincoln, where he practiced law, served as an officer of conservative lobby American Freedom Coalition, and gave occasional interviews to the media. Following his death, he was praised on the Senate floor in a speech delivered by Strom Thurmond, a contemporary of Curtis's who was also elected to the Senate in 1954.