He was elected as a judge of the Ohio Supreme Court in 1972, defeating Republican Robert E. Leach. He was re-elected in 1974, defeating Republican Sheldon A. Taft. Celebrezze was an Ohio state court judge; he also served in the Ohio State Senate.
In 1986, Celebrezze brought a successful libel suit against The Plain Dealer of Cleveland, based on an article that alleged that Celebrezze's campaign had accepted contributions from groups with organized crime connections.
In 1986, Celebrezze was defeated in a bid for another term as chief justice by Thomas J. Moyer. The race between Celebrezze and Moyer was dirty, with Moyer's campaign accusing Celebrezze of having links to organized crime.
In 1994, Celebrezze ran for a seat on the Ohio Court of Appeals for the Eighth District and lost to common pleas court Judge Terrence O'Donnell.
Celebrezze was the son of Cleveland politician Frank D. Celebrezze I and the nephew of Kennedy administration cabinet member Anthony Celebrezze. His first cousin, Anthony J. Celebrezze Jr., was an elected official and candidate for Ohio governor. His brother, James P. Celebrezze, also served on the Ohio Supreme Court.