She died in Milan after a long illness, Italy's Ansa news agency reports.
Jonathan Newhouse, chairman of publishers Conde Naste International, described her as "one of the greatest editors who ever made a magazine".
Celebrities including Madonna, Kanye West and Victoria Beckham paid tribute to her.
In a post on Instagram, Madonna posted a photo of her with the caption: "RIP Franca Sozzani! A trail blazer and a True Rebel! You are loved and adored by so many! We will miss you."
US fashion model Kendall Jenner tweeted: "Rest peacefully, Franca Sozzani."
Sozzani took charge of Vogue Italia in 1988 and became a key figure on the global fashion scene.
In 2008, she produced an "all black" issue, featuring black models only. It became an international hit.
Once asked how to define style, she said: "I think people should buy more mirrors than clothes, to see themselves before going out... but I think that clothes need to correspond to one's own personality. That is style."
Sozzani was also a champion of humanitarian causes. She served as goodwill ambassador to fashion for the UN, supporting workers in the fashion world in Africa and Asia and helping to raise money to fight hunger.
In a tribute on the Vogue website, US Vogue editor Anna Wintour described Sozzani as "warm, clever, funny".
"She made everything she worked on appear effortless, regardless of whether it was an event for several hundred; a whirlwind trip to Africa to support the continent's emerging designers; or the creation of yet another newsworthy, provocative, and utterly spellbinding issue of Italian Vogue," she said.
Sozzani attended the Fashion Awards in London earlier this month where designer Tom Ford presented her with the Swarovski award for positive change.

Why? Because of compound interest.
When you save money, it earns interest.
The lump sum grows from interest being added every year or every month.
Interest added on top of that interest is known as "compound interest" - and means that the longer you save, the better off you are.
So people who leave it late - like Dave in the example above - have compounded their financial problems when it comes to cashing in their savings.
But it is worth remembering that pensions usually depend on the success of investments which, unlike savings, do not guarantee a set level of interest.

You've tried on a pair of jeans, quite liked them, but won't commit until you've visited three or four other shops. You then return to the first place and buy those jeans, having wasted the best part of an hour.
Palermo owner Maurizio Zamparini is that shopper and Beppe Iachini is that pair of jeans, having been re-hired on Monday, just three months after getting the boot.
Between 10 November and 15 February, the struggling Sicilian Serie A side brought in four other bosses before realising Iachini was their man all along.
"I return to Palermo with great enthusiasm for the good of the club, the team and the fans," said the returning coach.
Zamparini asked Palermo fans for forgiveness for "this month of chaos", adding that this had been because of his "wrong evaluation and misunderstanding" of Iachini.
Nowhere did the owner promise to give that trigger finger a rest though.
Palermo have failed to win any of their last four league matches and are 15th in Serie A, with 26 points from 25 games, four points clear of the drop zone.
Sacking and re-hiring managers in a single season is not unusual in Italy.
During the 2013-14 Serie A campaign, Catania ditched Rolando Maran who then returned, only to be sacked again.
Meanwhile, in the same season, Livorno dismissed Davide Nicola - the coach who led them to promotion the previous season.
Replacements Domenico Di Carlo and Attilio Perotti failed to make an impression, so Nicola was asked back again.
Franca Sozzani, editor-in-chief of Vogue Italia for 28 years and a champion of Italian fashion, has died at the age of 66.
[NEXT_CONCEPT]
Saving a small amount of money regularly from when you start work will leave you better off than saving a bigger amount in later life.
[NEXT_CONCEPT]
We've all been there.