Allen died at his home in Hollywood on Sunday, two days before his 76th birthday, said family spokesman Mickey Cottrell. The specific cause was not given.
Allen had Parkinson's disease for the last two decades, Cottrell said, but he remained active directing plays until a few years ago.
Allen's death came a month after that of another "Rebel" alumnus, Dennis Hopper, who played one of the high school gang members.
A 1954 graduate of the UCLA theater department, where he won a best actor award, Allen had a few TV credits and un-credited bit parts in movies when he was cast as the arrogant, leather-jacket-wearing Buzz Gunderson in "Rebel Without a Cause."
The 1955 film, directed by Nicholas Ray, starred Dean as Jim Stark, the troubled new kid at school who is menaced by Buzz and his pals.
A knife fight between Dean and Allen's characters at the Griffith Observatory ends with the chicken run challenge: to drive two stolen cars toward the edge of a seaside bluff at high speed, and the first one to jump out of his car before it sails over the edge is a chicken.
"Why do we do this?" Jim asks Buzz that night at the bluff as they peer at the ocean below.
"You got to do something, don't you?" Buzz coolly replies.
The film, whose cast included Natalie Wood and Sal Mineo, opened the month after the 24-year-old Dean was killed in a car crash on Sept. 30, 1955.
Frank Mazzola, who played Crunch, one of the gang members who goes after Dean's character after Buzz is killed, recalled that he and Allen were "at odds" with each other when the cast made a screen test for the film.
"Corey was kind of an arrogant guy, and I was kind of teed off at him," Mazzola, who became a film editor, told The Times on Tuesday. "But then, because Nick was always trying to keep us together instead of hassling each other, it eventually resolved itself, and we became good friends.
"Corey was a good guy; they were all good guys. It was just a combination of all these kids that came together and made something special."
"Rebel Without a Cause" was the high point of Allen's screen-acting career.
Allen, who was born June 29, 1934, in Cleveland, went on to appear in films such as "Darby's Rangers," "Juvenile Jungle," "Party Girl," "Sweet Bird of Youth" and "The Chapman Report." He also made guest appearances on TV series such as "Perry Mason," "Bonanza" and "Dr. Kildare."
Allen was heavily involved in theater in Los Angeles, including being a partner in a touring theater operation launched in 1959 known as Freeway Circuit Inc., and co-founding a small repertory theater called Actors Theater in 1965.
He was already a theater director when he began directing in television in 1969. Over the next 25 years, he directed TV movies and numerous episodes of series such as " Hawaii Five-O," "Mannix," "The Streets of San Francisco," "Police Woman" and "The Rockford Files."
In 1984, he won an Emmy for outstanding directing in a drama series for an episode of "Hill Street Blues."
Allen later directed the two-hour TV pilot for "Star Trek: The Next Generation" and was one of the people involved in casting all of the regulars; he also directed other episodes of the 1987-94 series, as well as a number of episodes of the 1993-99 series "Star Trek: Deep Space Nine."
"Corey was a wonderful director and was adored by the actors," said Rick Berman, one of the executive producers of the "Star Trek" franchise.
Allen also taught acting at the Actors Workshop and the Margie Haber Studio, and he helped create the acting and directing curricula for Columbia College Hollywood.
He is survived by his daughter, Robin Duncan; his brother, Steve Cohen; four grandchildren and two great-grandchildren.
A memorial service will be held July 18. For details, call (818) 409-0448.