Albertini, born in Besana in Brianza, province of Monza e Brianza near Milan, emerged as a product of A.C. Milan's youth system, and went on to spend 14 highly successful years with the senior club after debuting in Serie A as a 17-year-old during the 1988–89 season under Arrigo Sacchi, on 15 January 1989, in a 4–0 home win over Como. He spent part of the 1990–91 season on loan at Padova Calcio in Serie B, collecting 28 appearances and 5 goals, in order to gain experience, and was subsequently awarded a prize by Diadora as one of the most promising young Italian stars. After a successful season with Padova, he soon established himself in the starting lineup of the Milan senior side during the 1991–92 season under Fabio Capello, wearing the number 4 shirt, and helping Milan to win the title undefeated that season; he would go on to make almost 300 Serie A appearances for the club (293 in total, scoring 21 goals), and 406 total career appearances for Milan, scoring 28 goals in all competitions.