One of his first film scores was for the 1973 film Executive Action, which put forward a conspiracy theory concerning the assassination of John Fitzgerald Kennedy in 1963. In the mid-1980s, Edelman wrote the theme to, and scored many episodes of MacGyver, a popular television series starring Richard Dean Anderson.

Edelman has been married to singer Jackie DeShannon since June 3, 1976. DeShannon is known for such 1960s hits as "When You Walk in the Room", "Put a Little Love in Your Heart" and "What the World Needs Now Is Love". Edelman is DeShannon's second husband and they have a son, Noah D. Edelman.

During the 1980s and early 1990s, he also collaborated with Ivan Reitman, producing scores for several of his comedies, including Ghostbusters II, Twins and Kindergarten Cop. He also contributed to Beethoven; The Last of the Mohicans; The Mask; Daylight; Anaconda; XXX; Gettysburg; My Cousin Vinny; While You Were Sleeping; Dragonheart; Shanghai Noon; Six Days, Seven Nights; The Indian in the Cupboard; Billy Madison; Angels in the Outfield; and EDtv to name just a few.

His music from The Adventures of Brisco County, Jr. is featured during NBC's Olympic Games coverage when announcing upcoming events. In addition, it was used during NBC's coverage of the 1997 World Series. At the end of the 1996 Summer Olympics, it used the closing music of Gettysburg. In the 1990s, Edelman composed the popular theme music for NBC's NFL telecasts which was used for the 1995–97 seasons through Super Bowl XXXII.

Edelman was honored with the Richard Kirk Award at the 2003 BMI Film and TV Awards. The award is given annually to a Composer who has made significant contributions to film and television music. In 2004, he received an honorary doctorate in fine arts from the University of Cincinnati. He was given the degree along with three other honorees distinguished in various other fields, including Coretta Scott King.

He produced the scores for the 2008 film The Mummy: Tomb of the Dragon Emperor. This score was recorded at Abbey Road Studios in London, where in 2010 Randy also recorded and composed the Irish flavored music score for Amy Adams film Leap Year. He was awarded the Goldspirit Award (named in honor of Jerry Goldsmith) for best comedy score of 2011 for the Leap Year Soundtrack CD on Varese Sarabande.