Enrico Cucchi made his professional debut in 1981 with Savona in Serie C2, which was coached by his father Piero at the time. In 1982, he was sold to Inter and he made his Serie A debut with the club three years later, on 13 January 1985, in a 1–1 away draw against Ascoli. He became a permanent first-team member the subsequent season, making 22 appearances, and scoring a goal against Lecce, also making his European debut with the club in the UEFA Cup. After the 1986–87 season, he spent the next two seasons on loan at Empoli and Fiorentina, before returning to Inter again in 1989. During the 1989–90 season, he scored in Inter's 2–0 victory over Sampdoria in the 1989 Supercoppa Italiana. He was sold to Bari the following season, where his career declined due to illness. He was operated on his thigh due to a growing mole, which was later revealed to be melanoma. He spent his final season with Ravenna during the 1993–94 Serie B season, making only 5 appearances due to his illness, before retiring in 1995.