The Swiss were two minutes away from taking their match against the South Americans into a penalty shootout before Angel di Maria scored a late winner.
"Pride prevails today," he told reporters. "The team showed a passionate performance and we were tactically very mature.
"We managed to stay calm. We saw everything that can happen in the life of the coach: the shock of the goal against us and our incredible reaction to go for it all. This is something that will make us proud."
Following the match, Hitzfeld has announced his retirement from football management.