Curci lead the University of Tampa Spartans to a 25–6 record in three seasons (1968–1970). After his team defeated the Miami Hurricanes at the Orange Bowl in 1970, and Tampa finished that season 10–1, he was hired by the University of Miami.
Curci's record at Miami was 9–13. He was head coach at Miami during the infamous Florida Flop in 1971, when the Florida Gators defense allowed Miami to score a touchdown late in the fourth quarter by dropping to the ground mid-play. The defense wanted quarterback John Reaves to get the ball back and set an NCAA career passing record. Florida won the game 45–16. Curci was particularly upset at the Gators' actions, since the game was well out of reach by then. He refused to shake hands with Florida coach Doug Dickey after the game, and was quoted as saying, "I lost all respect for [Dickey] as a coach and as a man. What he did shows no class... I think he made a fool of himself."
During his tenure at the University of Kentucky, Curci compiled a record of 47–51–2. The 1976 Kentucky Wildcats finished 9–3 and 5–1 in conference play—their first winning season in 13 years—and defeated North Carolina in the Peach Bowl 21–0, finishing ranked #18 in the final Associated Press poll. After that season, the Wildcats were slapped with two years' probation for numerous recruiting and amateurism violations. The Wildcats were also banned from bowl games and live television in 1977, and limited to only 25 scholarships in 1977 and 1978. The 1977 team finished 10–1, including a win at #4 Penn State and an undefeated record in conference play though they were ineligible for the title due to probation. They finished the season ranked #6 in the final Associated Press poll. It was only the second 10-win season in school history.
Due in part to the loss of scholarships from the 1976 infractions case, Curci was never able to put together another winning team. In his last four years, he only won eight games in SEC play. Curci's tenure as Kentucky's coach ended after nine seasons, longest of any Kentucky coach. In his final game as Kentucky's coach, Curci led the Wildcats to a 21–10 victory over Tennessee on November 21, 1981.
Curci later coached in the Arena Football League with the Tampa Bay Storm in 1991, the Cincinnati Rockers in 1992. He led the Storm, quarterbacked by current Washington Redskins head coach Jay Gruden, to an ArenaBowl V championship and was named Coach of the Year in his only season with Tampa Bay. Curci also did radio broadcasts for Tampa Bay Buccaneers and college football games.