Sciarra was an All-American quarterback at UCLA. He starred in the 1976 Rose Bowl in which UCLA upset favored Ohio State University, which was unbeaten ranked number one in the nation, by a score of 23–10. Sciarra was named Most Valuable Player in the game. He ran for the most career yards by a quarterback in UCLA history.
The Tampa Bay Buccaneers signed Sciarra, and the Philadelphia Eagles traded an undisclosed draft pick to Tamp Bay in exchange for Sciarra in 1978. While with the Eagles, he played for coach Dick Vermeil, who had been his college coach at UCLA. The Eagles were a successful team at that time and won the NFC Championship in 1980 and played in Super Bowl XV. He was re-signed to a two-year contract on June 15, 1981, and played with the Eagles until 1983.
On August 8, 1991, Sciarra was inducted into the Rose Bowl Hall of Fame. He will be inducted into the College Football Hall of Fame as a member of the Class of 2014.
Sciarra was the keynote speaker at UCLA's 2009 Sociology Department graduation ceremony, where graduates included Hall of Fame quarterback Troy Aikman, then-current quarterback Kevin Craft, and former linebacker Marcus Reece, who had come back to UCLA to finish his degree.