As a youngster, David Barcéna showed promise as a swimmer but after training as a cadet in the Mexican Army, he graduated as a second lieutenant in 1963 and by then he had turned his sporting attention to the multi-discipline modern pentathlon and competed in the first of five consecutive Olympic Games at Tokyo in 1964. He again took part in the pentathlon on home soil four years later before turning to three-day eventing at the 1972 and 1976 Olympics. After four Games without a podium finish he eventually won that elusive medal, a bronze, in the eventing team competition at Moscow 1980. This medal came five years after his first international success when he won a bronze medal in both the individual and team eventing competition at the 1975 Pan American Games, where 1980 team-mate Manuel Mendívil was also in the Mexican medal-winning team. Barcéna took part in the 1963, 1966 and 1967 Modern Pentathlon World Championships, winning the riding element in the latter two years.
David Barcéna twice received a first class Mexican sports honour, and in 1980 won the Mexican National Sports Award. He spent most of his life in the Army and served as commander of the military attachment in the Mexican city of Delicias, where he continued to live after his retirement, and where he died in 2017.