Astori was found dead in his hotel room ahead of his side's Serie A clash with Udinese, with an autopsy later confirming he died of heart problems owing to natural causes.
The defender started his career with the Milan youth academy before signing for Cagliari in 2008, after failing to break into the Rossoneri first team.
But Gattuso, who spent 13 years at San Siro as a player, still remembers Astori as a polite and humble man who wished him well after his appointment as head coach last November.
"Astori was a great player on and off the pitch," he said. "He is a great example for all young men. I hope we never forget him. It's important to remember men like him.
"I was at Milan when he was in the youth teams. Whenever I saw him in the gym, he would ask me if he could use it. The third time that happened I told him that it was his home — he didn't need my permission.
"The last time I saw him was on December 30 in Florence. Before the match, he said he was proud and happy for me and my new adventure with Milan.
"He was always respectful and had a fantastic attitude. The team was very shaken on Sunday. There was none of the usual music in the dressing room."
Gattuso's side hosts Arsenal in the first leg of their UEFA Europa League Round of 16 tie on Friday (AEDT), looking to extend its 13-game unbeaten run in all competitions.
The 40-year-old feels the Gunners should still be considered favourites given their squad depth and resources, despite them having lost their last four matches, but he believes Italian sides deserve credit for helping youngsters to flourish.
"The Italian teams are doing well in Europe, but we can't compare English football with Italian football nowadays," he said. "Ten years ago, it was the opposite, but today the top players go to the Premier League.
"At the same time, this has been an opportunity for our young players. It's not a coincidence that today we speak about [Patrick] Cutrone and [Davide] Calabria. Because of these changes, they have been able to show their qualities, like many other young Italians."