Graziani was born in Subiaco, in the province of Rome. A prolific and physical striker, he started his footballing career in Bettini Quadraro before moving to Arezzo and then to Torino in 1973. Graziani played eight seasons for Torino, making his debut in Serie A on 18 November 1973 against Sampdoria and scoring his first goal in the top flight on 16 December of that same year against Bologna. In total, Graziani scored 122 goals in 289 games for Torino, divided as follows: 221 appearances (97 goals) in the league, 45 appearances (17 goals) in the Coppa Italia and 23 appearances (8 goals) in European competition. He won the "Scudetto" in 1975–76.