Minkoff was involved in theater since age ten and appeared in numerous productions for the Palo Alto Children's Theater and Theater Works. He decided to work for Disney after encountering art historian Christopher Finch's landmark study, "The Art of Walt Disney", while babysitting as a teen. Coincidentally the children he was minding were Finch's nieces and were mentioned in the book's dedication. (Minkoff would finally meet and work with Finch on the author's later book, "The Art of The Lion King"). Minkoff began his formal training as a student at CalArts majoring in character animation. He served a fateful summer internship at Disney in 1982 which allowed him to work with one of the company's revered "Nine Old Men", veteran animator Eric Larson, who helped him prepare a personal animation test. This led to Minkoff being hired by the feature animation department in 1983, making his feature debut as an in-betweener on the animated flop, "The Black Cauldron" (1985).
Minkoff proved to be a fast learner. He designed several characters for "The Great Mouse Detective" (1986) including the hero, the Sherlockian Basil of Baker Street. By the time the production was over, Minkoff had advanced to supervising animator. Subsequent credits include co-writing a song for "Oliver & Company" (1988), character animation for "The Little Mermaid" (1989), and script development for "Beauty and the Beast". In addition to his duties on Roger Rabbit cartoons, Minkoff has helmed several projects for the Disney MGM Studio attraction in Florida.