Minghella will chair the BFI, which promotes film education and heritage, for three years.
He replaces Joan Bakewell who has led the BFI since 1999.
Minghella, - whose films include The English Patient and The Talented Mr Ripley - said: "As a film-maker and film-goer I have long enjoyed the resources and facilities provided by the BFI.
"The BFI exists to celebrate the art of film by preserving its past and by providing access to the treasure trove of world cinema.
Archive
"I believe there is a good deal to be said for having a practising film-maker at the centre of the BFI's policy and future and I will do my best to listen and learn from those working inside the BFI and from those who contribute to its activities.
"Most of all, I want to promote a BFI whose work is better understood by the public and more accessible to it."
The BFI is charged with looking after the UK's film and television archive, and it also runs the London Film Festival and the capital's National Film Theatre.
Sir Alan Parker, who chairs the government-backed Film Council which made the appointment, said:
"Anthony Minghella is one of the world's greatest film writers and directors. His achievements are truly inspirational, as is his passion for the art of film."
Minghella has recently been working on his new film, romantic drama Cold Mountain, which stars Jude Law, Nicole Kidman and Renee Zellweger.