Minkoff was born in Palo Alto, California. He studied at Palo Alto High School and graduated from California Institute of the Arts in the early 1980s in the Character Animation department. While he was working his way through CalArts, he was hired by Disney as a cleanup artist and worked on "The Black Cauldron".
He has directed several films for Walt Disney Feature Animation, including The Lion King (1994) and two of the Roger Rabbit shorts: Tummy Trouble (1989) and Roller Coaster Rabbit (1990). While working at Disney he wrote the song "Good Company" for Oliver & Company. He also made the film The Haunted Mansion (2003).