Grusin's mother was a Pianist and his father was a Violinist from Riga, Latvia. An alumnus of the University of Colorado at Boulder, College of Music, he was awarded his bachelor's degree in 1956. His teachers included Cecil Effinger and Wayne Scott, Pianist, arranger and professor of jazz.

Grusin composed theme music for the TV programs It Takes a Thief (1968), The Name of the Game (1968), Dan August (1970), The Sandy Duncan Show (1971–72), Maude (1972), Good Times (1974), Baretta (1975), St. Elsewhere (1982), and, for Televisa in Mexico, Tres Generaciones (1987). He also composed music for individual episodes of each of those shows. His other TV credits include The Wild Wild West (1966), The Girl from U.N.C.L.E. (1966), and Columbo: Prescription: Murder (1968). He also did the theme song for One Life to Live (1968) from 1984–92. Grusin and Larry Rosen founded GRP Records in 1978. In 1994, GRP was in charge of MCA's jazz operations. Founders Grusin and Rosen left in 1995 and were replaced by Tommy LiPuma. In 1997, Grusin and Rosen founded N2K Encoded Music, which was renamed N-Coded Music.

In the late 1970s, he started GRP Records with his Business partner, Larry Rosen, and began to create some of the first commercial digital recordings. He was the Composer for The Graduate, On Golden Pond (1981), Tootsie (1982) and The Goonies (1985). In 1988, he won the Oscar for best original score for The Milagro Beanfield War, He also composed the musical scores for the 1984 TriStar Pictures and the 1993 Columbia Pictures Television logos.

He received honorary doctorates from Berklee College of Music in 1988 and University of Colorado, College of Music in 1989. Grusin was initiated into the Beta Chi Chapter of Phi Mu Alpha Sinfonia at the University of Colorado in 1953.

From 2000-11, Grusin concentrated on composing classical and jazz compositions, touring and recording with collaborators, including Guitarist Lee Ritenour. Their album Harlequin won a Grammy Award in 1985. Their classical crossover albums, Two Worlds and Amparo, were nominated for Grammys.