Graziani was born in Subiaco, in the province of Rome.
As striker, he started his footballing career in Arezzo. Later he became famous as one of the leading members of the last scudetto won by Torino. He joined the granata in 1973, and soon became a key player for the team, winning the Serie A in 1976. In 1977, he was the Serie A top-scorer with 21 goals.
Graziani then played for ACF Fiorentina (1981-1983) and AS Roma (1983-1988), and failing to win a Champions League in a penalty defeat against Liverpool (Graziani himself missed a penalty shoot-out in the match). He then played also for Udinese, before abandoning his playing career. Graziani totalled 353 appearances, with 130 goals, in the Italian Serie A.
He was also an international player for Italy, representing the Azzurri at the 1978 FIFA World Cup, 1982 FIFA World Cup and in Euro 80. He scored one goal in the 1982 FIFA World Cup against Cameroon, a tournament that Italy won. With 23 goals in 64 caps, he is (as of April, 2006) ranked as the ninth highest all-time scorer for his national team.
Graziani's son, Gabriele, is a football player too; he currently plays in the role of forward for Serie C1 team Cremonese.