Byrne won the U.S. Open Chess Championship in 1953 in Milwaukee and around that time he achieved the second-highest rating in the U.S., behind Samuel Reshevsky, against whom Byrne had a winning record. He was awarded the International Master title by FIDE (the World Chess Federation) in 1962, and played for or captained five U.S. Chess Olympiad teams between 1962 and 1972. In 1972, he led a team representing Pennsylvania State University (the remainder of the team was alumni) to the US Amateur Team Championship in Philadelphia. The winning Penn State team consisted of Byrne, Dan Heisman, Steve Wexler, Bill Bickham, and Jim Joachim (alt.). Byrne's elder brother, Grandmaster Robert Byrne, was also a leading player of that time.