When he was 12-years-old Tristán Falfán stepped into a boxing ring for the first time, participating in neighbourhood amateur championships in Córdoba, Argentina. By 1956 Falfán had become the Argentine champion and as a result qualified for the Melbourne Olympics. At the Melbourne Games he defeated Maurice White, after the referee stopped the contest, before losing to Henryk Niedźwiedzki in the quarter-finals. Falfán turned professional in 1957, fighting in 55 bouts, winning 38 of them, including 19 by knock-out.