Andrews was born in Cass County, North Dakota. He attended the public schools there. In 1944 at the age of 18, Andrews was admitted to the United States Military Academy. He quit in 1946 after receiving a disability discharge. He then attended the University of North Dakota at Grand Forks, North Dakota and graduated in 1949.

Andrews then became a farmer. During the 1950s he began to enter politics, serving on farmers' organizations and Republican committees. In 1963 Andrews became the Republican candidate for a seat in the United States House of Representatives from North Dakota when a special election was required after the death of congressman Hjalmar Nygaard. Andrews won the election. He was reelected to a full term in 1964 and served in the House until 1981, being reelected every two years.

In 1980 Andrews decided not to run for reelection to the House. Instead, he ran for one of North Dakota's seats in the United States Senate which was being vacated by the long-serving Republican Milton Young who announced that he would not run for reelection and retire after the election. Andrews won the election and served in the Senate for one term, from 1981 to 1987. He was chairman of the select committee on Indian affairs from 1983 to 1987.

Andrews was defeated for reelection in 1986 by Democrat Kent Conrad. Andrews retired from elective politics. He started a consultancy firm in Washington, D.C. but currently lives in Mapleton, North Dakota. Grand Forks International Airport in Grand Forks, North Dakota has sometimes been called Mark Andrews International Airport after him, but usage of the name has dropped out of popularity.