Vialli, who died on Friday morning aged 58, was the first Italian to manage in the Premier League when he was appointed player-coach at Chelsea in 1998.
Spurs head coach Antonio Conte, a friend and former teammate of Vialli's at Juventus, learned of the news on his way to work and was too upset to take his press conference ahead of Saturday's FA Cup third-round tie against Portsmouth, with Stellini stepping in.
"He opened the door for Italian managers," Stellini said of Vialli. "He was the first [in England]. In that moment, maybe the Premier League for us was different, not a country for Italian managers. But in that moment the Premier League grew a lot and overcame the Serie A.
"Now we have to say thanks to Vialli for opening this door to everyone. He let us understand how important football in Europe is, to open doors. Because an Italian manager came into the Premier League and improved the Premier League together. It's a great thing."
Vialli and Conte won the Champions League together at Juventus in 1996 and were also teammates for the Italian national team.
"Antonio in this moment is upset and sad and every one of us we are close to Vialli's family and all the people that love Vialli," Stellini continued. "He's an important person in Italy and also in England. We're close together to stay with his family and remember him in the right way.
"He was a great player but first of all was a great man. He teach us a lot of things, not only when he played but when he spoke with everyone. When you know him, and the people who know him, can only speak well about this great guy. He was a great leader for Italy."