Born in Cali of Lebanese descent, Mondragón began his career began at Deportivo Cali, and he moved to Independiente Santa Fe and then Cerro Porteño in Paraguay. His career took off when he joined the Argentine first division side Argentinos Juniors in 1993, and then made his international debut for Colombia. The following year he moved to Club Atlético Independiente where he won, among other titles, the 1995 Supercopa Libertadores against Flamengo. Apart from some brief periods, he largely stood at the club until 2001.
After making his debut against Venezuela in 1993, Mondragón was a member of the Colombian national teams that competed at the 1992 Summer Olympics and the 1994 and 1998 FIFA World Cup. During the 1998 World Cup, he started in goal for all three of their games, including the final match against England. Despite conceding two goals, he made some impressive saves and in doing so kept the score down to 2–0, with the BBC's South American football correspondent Tim Vickery saying that Mondragón was "single-handedly responsible for the fact that England did not run up a cricket score". At the end of the game, Mondragón broke down in tears and David Seaman, England's goalkeeper, did his best to console him. According to German footballing legend Franz Beckenbauer, Mondragón had been the best goalkeeper of the first round.