The 39 year-old midfielder is preparing for a first season in Serie A after his summer arrival from Real Madrid.
Costacurta told La Gazzetta dello Sport: "Still decisive at his age? Someone as successful as him, with that spirit and desire, can always be. It's no coincidence that two hip movements were enough to make AC Milan fans fall in love. It took very little to understand what he can offer.
"San Siro is a stadium unlike any other, it's exciting even if you've played in Madrid for years. And he was; I saw him engaged and impressed by the welcome. Playing once a week could help his performances. On a personal level, I trust (Zvonimir) Boban, who for thirty years has spoken of Luka as an exceptional person as well as a fantastic champion."
Meanwhile, on Milan's title chances, Costacurta was adamant.
He insisted: "It absolutely must be a goal. The Champions League will be decisive and tiring for the other teams. Milan isn't the favorite, but by playing once a week they can join the title race alongside Napoli and Inter.
"It's no coincidence that (Rafael) Leao already has a different face: in the first few games last year, the team wasn't so focused. Max (Allegri) brings something different, even in terms of energy. We'll wait a few more games to better understand whether Allegri's turnaround can be so clear and incisive right away."