La Viola's top-flight match with Atalanta BC on Sunday was postponed after Barone was taken ill at the team hotel in front of players and was subsequently rushed to hospital.
Fiorentina confirmed that the club chief had suffered a cardiac arrest and was in intensive care at the San Raffaele Hospital in Milan, where "artificial mechanical support techniques" were supporting his vital functions.
On Tuesday afternoon, Fiorentina released a new statement announcing that Barone had lost his life one day before his 58th birthday, saying: "It is with deep sorrow and immense sadness that Fiorentina confirms the loss of one of its figureheads, a person who has marked the recent history of the club and who will never be forgotten.
"General Director Giuseppe Barone passed away today at the San Raffaele Hospital in Milan after suddenly falling ill on Sunday.
Rocco Commisso and his family, Daniele Prade, Nicolas Burdisso, Alessandro Ferrari, Vincenzo Italiano, Cristiano Biraghi and everyone associated with Fiorentina are utterly devastated by the terrible loss of a man who showed great professionalism and gave his heart and soul for these colours.
"He was a treasured friend who always stayed strong in the happiest and, above all, the most difficult moments. The entire Viola world sends its heartfelt condolences to Giuseppe's wife Camilla, his children and the entire Barone family during this time of immense sorrow."
Barone's death comes almost exactly six years after former Fiorentina captain Davide Astori died aged just 31, and club captain Biraghi - who was in his first season with the Viola when Astori died in 2018 - paid a moving tribute to Barone on social media.
"Ciao Joe, unfortunately you too played a nasty trick on me by going without saying goodbye! I cannot fathom how this wonderful life can also be so cruel and cutting," the defender wrote.
"I struggle to find the words, or rather I'd have too many, to salute you the way you deserve, but I just want to remember how we met. You and me, two strong and blunt characters who from the first meeting crashed like two trains at full speed, but with the fairness that stands us apart, in the end were travelling in the same direction, against everyone and everything else!
"In recent years our rapport had become so strong that we would shield each other from the great deal of criticism that we received, you and I! You were a friend, a big brother or a father maybe, a central person in my life since I've known you! One of the very few who were always by my side and you had no shame in saying it!
"This is why I tell you that I will not forget the promise that we made each other. Make sure you don't forget it either, so from up there give me the strength to keep it going! You and I know! I love you and I will do everything to make true the dreams that you had for our Fiorentina! It was a tough blow for me and the squad, especially as you left without even giving us a warning!
"But we forgive you because all that you did for us will be priceless for life! I'll say goodbye now, because I have to keep our mission going! Ciao Joe. Your Cap."
Barone had served as Fiorentina's Chief Executive Officer since 2019, and during his five years in office, La Viola finished as runners-up to West Ham United in the 2022-23 Europa Conference League final and achieved four top-half finishes in Serie A.
Italiano's side are also in the semi-finals of this year's Coppa Italia - preparing to take on Atalanta BC over two legs next month - either side of a double-header with Viktoria Plzen in the Europa Conference League quarter-finals.
La Viola sit eighth in the Serie A standings and are due to return to action against AC Milan on March 30 following the international break.