Minghella's publicist, Jonathan Rutter, said the filmmaker died at London's Charing Cross Hospital. He said Minghella was operated on last week for a growth in his neck, "and the operation seemed to have gone well. At 5 a.m. today he had a fatal hemorrhage."
Former British Prime Minister Tony Blair, who became friends with Minghella after the filmmaker directed a Labour Party election ad in 2005, said he was "really shocked and very sad."
"Anthony Minghella was a wonderful human being, creative and brilliant, but still humble, gentle and a joy to be with," Blair said. "Whatever I did with him, personally or professionally, left me with complete admiration for him, as a character and as an artist of the highest caliber."
"He was a brilliantly talented writer and director who wrote dialogue that was a joy to speak and then put it onto the screen in a way that always looked effortless. He made work feel like fun. He was a sweet, warm, bright and funny man who was interested in everything from football to opera, films, music, literature, people and most of all his family."
The 1996 World War II drama "The English Patient" won nine Academy Awards, including best picture, best director for Minghella and best supporting actress for Juliette Binoche. Based on the celebrated novel by Canadian writer Michael Ondaatje, the movie tells of a burn victim's tortured recollections of his misdeeds in time of war.
In a 1996 interview with The Associated Press, Minghella said too many modern films let the audience be passive, as if they were saying, "We're going to rock you and thrill you. We'll do everything for you."
"('The English Patient') goes absolutely against that grain," he said. "It says, `I'm sorry, but you're going to have to make some connections. There are some puzzles here. The story will constantly rethread itself and it will be elliptical, but there are enormous rewards in that."'
His 2003 "Cold Mountain," based on Charles Frazier's novel about the U.S. Civil War, earned a best supporting actress Oscar for Renee Zellweger.
The 1999 "The Talented Mr. Ripley," starring Matt Damon as a murderous social climber, was based on a novel by Patricia Highsmith. It earned five Oscar nominations.
Among his other films were "Truly, Madly, Deeply" (1990), and last year's Oscar-nominated "Michael Clayton," on which he was executive producer.
Minghella also turned his talents to opera. In 2005, he directed a highly successful staging of Puccini's "Madama Butterfly" at the English National Opera in London choreographed by Minghella's wife, Carolyn Choa. The following year, he staged it for the season opener of New York's Metropolitan Opera. It was the first performance of the Met's new era under general manager Peter Gelb.
Minghella was recently in Botswana filming an adaptation of Alexander McCall Smith's novel "The No. 1 Ladies' Detective Agency." Due to air on British television this week, the book is the first in a series about the adventures of Botswanan private eye Precious Ramotswe; a 13-part television series was recently commission by HBO.
Jeff Ramsay, press secretary to Botswanan President Festus Mogae, said the director had been coming to the country ahead of the detective film and learning about Botswana. He said Minghella had told him how he had been forced to shoot "Cold Mountain" in Romania and that it had "seemed wrong." He said this made the director "more sure that the film could only be shot in Botswana."
"Much more than a TRUE artist who thrilled in offering his divine gift with the world, Anthony Minghella was a dear, dear trusted friend," Jill Scott, who plays Ramotswe, said in a statement. "My heart aches with grief. Words can not express how deeply he will be missed or how deeply he was loved."
Born the second of five children to southern Italian emigrants, Minghella came to moviemaking from a flourishing playwriting career on the London "fringe" and, in 1986, on the West End with the play, "Made in Bangkok," a hard-hitting look at the sexual mores of a British tour group in Thailand.
He worked as a television script editor before making his directing debut with "Truly, Madly, Deeply," a comedy about love and grief starring Juliet Stevenson and Alan Rickman.
Producer David Puttnam told the BBC that Minghella was "a very special person."
"He wasn't just a writer, or a writer-director, he was someone who was very well-known and very well-loved within the film community," Puttnam said. "Frankly he was far too young to have gone."
Minghella is survived by his wife; his actor son, Max Minghella; and his daughter, Hannah.
By Jill Lawless