 Alessandro Costacurta aizivikanwa nevatsigiri vake se Billy, nekuda kwechimiro chake chakanaka muhuduku hwake, uye unyanzvi hwake hunoshamisa mu basketball sezvikwata zvebhasikiti zveMilan, Olimpia Milano, zvainzi Billy munguva yekupedzisira ye1970s, mutsigiri mukuru wechikwata. Pakutanga chigadzirwa che AC Milan vechidiki, panguva yebasa rake rehunyanzvi Costacurta aizoenderera mberi nekutamba kukirabhu yevakuru kweanopfuura mwaka makumi maviri, mushure mekupfupika kwechikwereti kuMonza muSerie C1 mumwaka wa1986-87. Costacurta akanga atova nhengo yechidiki yechikwata chevakuru mumwaka wa198586, asi akakundikana kuita chitarisiko chimwe chete. Asati atumirwa kuMonza pachikwereti, Costacurta aizoita Milan yake yekutanga muCoppa Italia mumwaka wa1986-87, asi aisazoita Serie A yake yekutanga neMilan kusvika gore rinotevera, pasi paArrigo Sacchi. Costacurta akaita yake yekutanga Serie A kuoneka musi wa25 Gumiguru 1987 vachipesana neHellas Verona pazuva rechitanhatu remutambo wemwaka. Milan yaizoenderera mberi kuhwina mutambo 10, pamwe nemusoro weSerie A, asi Costacurta akagamuchira mikana shoma yekutamba mwaka iwoyo, nekuda kwekuvapo kweane ruzivo Filippo Galli pachinzvimbo chake. Pakazara, Costacurta akabata 7 kuoneka muSerie A, uye kuoneka kumwe muCoppa Italia mwaka iwoyo. 