Born in Roswell, New Mexico, Burciaga received a B.S. from the U.S. Military Academy in 1952 and was a U.S. Air Force Pilot from 1952 to 1959. He received a J.D. from the University of New Mexico School of Law in 1963. He was a law clerk, Hon. H. Vearle Payne, U.S. District Court, New Mexico from 1963 to 1964. He was in private practice in Albuquerque, New Mexico from 1964 to 1968. He was a special assistant state attorney general of Office of the State Attorney General in Santa Fe, New Mexico from 1967 to 1969, thereafter returning to his private practice from 1969 to 1979. He was also a lecturer at the University of New Mexico School of Law from 1970 to 1971, and was a special prosecutor for the First Judicial District in Santa Fe in 1975.

On July 19, 1979, Burciaga was nominated by President Jimmy Carter to a new seat on the United States District Court for the District of New Mexico created by 92 Stat. 1629. He was confirmed by the United States Senate on October 31, 1979, and received his commission on November 2, 1979. He served as chief judge from 1989 to 1994, assuming senior status on November 9, 1994. He ruled against The NCAA in a case that allowed schools to contract television rights for their football programs thus leading to the many games seen now. Burciaga served until his death, in Albuquerque.