During his playing career, Bauza spent most of his career playing for Rosario Central from 1977 to 1982, where he found his most successful achievement in club football. He won two Primera División and scored 80 league goals in 310 league appearances for Rosario Central, making him one of the highest scoring defenders in the history of Argentine football.
In 1983, he transferred to Atlético Junior and played for them for two years. Later on in his career, he also played for Independiente in 1985–86 but he only got seven appearances. He returned to Rosario Central in 1986.
Between 2003 and 2004 he spent some time working as a TV pundit before attempting to resurrect his managerial career with Sporting Cristal of Peru. Within six months of joining the club, he had led them to the Peruvian championship. He remained with the club until 2005 when he left after a downturn in results.
After returning to Colón for a season he took over as manager of LDU Quito of Ecuador in mid-2006, leading them to a league championship in his second season.