Born in Athens, Texas, Griffin attended Texas A&M University, where he studied aeronautical engineering. He joined NASA in 1960 and became a flight controller, specializing in guidance and navigation systems. In 1967, he became a flight director. He was lead flight director for Apollo 12, Apollo 15, and Apollo 17, as well as playing an important role in the resolution of the Apollo 13 crisis.
From August 1982 until January 1986, Griffin served as director of the Johnson Space Center in Houston, Texas.
Griffin was a technical adviser for the films Apollo 13, Contact, and Deep Impact. In the HBO miniseries From the Earth to the Moon, he was played by actor David Clyde Carr.