Born in Pontremoli, at club youth level, Albertosi represented the Spezia youth academy for a season, before making his debut in Serie A at the age of 18 for Fiorentina on 18 January 1959, playing in a goalless draw with A.S. Roma. Despite heavy competition from Giuliano Sarti during his first five seasons at the Tuscan club, where he initially mainly served as a back-up, Albertosi represented Fiorentina until 1968, winning a Mitropa Cup in 1966, a European Cup Winners' Cup in 1961 (the first ever edition of the competition), and 2 Coppa Italia trophies in 1961 and 1966. He later transferred to Luigi Riva's side, Cagliari Calcio, where he helped to win Cagliari's legendary first and only Scudetto during the 1969–70 season, his second season with the Sardinian club, recording what was at the time the lowest number of goals conceded in a 16-team Serie A campaign. He later transferred to A.C. Milan in 1974, where he remained for six years, and he won yet another Serie A championship medal with the club during the 1978–79 season, along with another Coppa Italia during the 1976–77 season. With 233 appearances for the club, he is Milan's fifth-most capped keeper of all time, behind only Christian Abbiati (380), Sebastiano Rossi (330), Dida (302), and Lorenzo Buffon (300).