Missoni co-founded the fashion brand in 1953 with his wife Rosita Jelmini and their designs have graced the rich and famous from Jackie Kennedy to the Duchess of Cambridge Kate Middleton.
He had been hospitalized with heart trouble last week and died peacefully surrounded by his family in his villa in Sumirago in northwest Italy, where he and the company have been based for decades.
"The rainbow that sprang from his creations gave us the impression of a happy man who managed to bring his fashion to a global audience," said Milan mayor Giuliano Pisapia.
"The jumpers and dresses he created have given him eternity," he said, adding that Missoni had "rendered great the Made in Italy brand".
The family said there will be a lying-in state on Sunday in the courtyard of his textile factory in Sumirago and the funeral will be on Monday.
The Italian fashion chamber expressed its condolences and said that luxury boutiques on Milan's famed Via Montenapoleone would open later than usual on Monday as a sign of respect.
'An innovative and creative genius'
"Missoni was one of the founders of Italian fashion and of high-end pret-a-porter, guiding his company to innumerable successes together with his wife Rosita," the fashion chamber, which organises Milan Fashion Week, said in a statement.
It said Missoni had been "an innovative and creative genius".
Missoni was born in the then kingdom of Yugoslavia in what is now Dubrovnik on February 11th, 1921 and after moving to Italy he initially began a career in track athletics - a sport he pursued into old age.
He became a national champion before the Second World War and took part in the 1948 London Olympics.
During the war, he fought in the Battle of El-Alamein and was held as a prisoner of war.
At the Olympics he met his future wife, whose family owned a textile business in northern Italy.
The Missoni brand quickly earned a reputation for testing new boundaries in the 1960s and was kicked out of the Pitti fashion shows in Florence when its models did not wear bras on the catwalk.
Missoni was a self-effacing, jovial man who told one interviewer that the geometric patterns on his dresses "were like that simply because we had machinery that could only make straight lines."
He also ascribed the patterns to the squares in the exercise books he used to design them.
But the company kept up a reputation for innovation in recent years and was the first to delve into the mass market through a successful partnership with US mega-store chain Target.
It also followed other major Italian fashion chains in setting up branded hotels in different cities including Edinburgh and Kuwait.
The company, which exports around 80 percent of its production, had a turnover of 150 million euros ($197 million) in 2011.
Missoni continued to work at the company until his death, although he had handed managerial responsibility to his two sons and daughter.
He suffered tragedy earlier this year when a plane carrying his eldest son, Vittorio, and five other people went missing on a flight from the Venezuelan island resort of Los Roques.
The 58-year-old is now feared dead and those who knew him best were quoted by Italian media saying his father had not been himself since, and had refused to talk about it with the family.