Nicknamed "l'Abatino", and "the Golden Boy" of Italian football throughout his career, Rivera was the product of his hometown club's youth football academy; he joined the first team in 1958 and made his debut in Serie A for the Alessandria senior side against Internazionale on 2 June 1959 at the age of only fifteen years, nine months and fifteen days, in a 1–1 draw; later that year, he scored his first Serie A goal in a 2–2 home draw with Sampdoria on 25 October. He is the third youngest player in history to debut in Serie A, behind only Amedeo Amadei and Pietro Pellegri, and the second youngest goalscorer in Serie A, behind only Amadei. A year later, he was bought by AC Milan on a co-ownership deal, as an eventual replacement for the club's captain Juan Schiaffino in the playmaking role; although the club's officials were concerned about Rivera's diminutive physique during his trial, Schiaffino insisted that the talented youngster be offered a contract, and later served as a mentor to the young attacking midfielder. Rivera spent his first season at the club back on a year-long loan with Alessandria; during the season, he scored a goal in a 1–1 draw with Milan on 7 February 1960, and played 26 matches in total, in which he scored 6 goals, also earning the nickname "Il Signorino" (the little gentleman). Despite Alessandria's relegation, Rivera's performances throughout the season impressed Milan, who later that year signed the youngster outright from Alessandria for another 65 million Lire, making his total transfer fee at the time a notable sum of 130 million Lire; the transfer fee paid by Milan to Alessandria also included the sale of Giancarlo Migliavacca, and Sergio Bettini on loan. Rivera's final appearance with Alessandria came on 19 June 1960, in a 2–0 home defeat against FC La Chaux-de-Fonds in the first ever edition of the Coppa delle Alpi.