Argentina, gold medal winners in their last two appearances at the Games in Athens and Beijing, fluffed the chance to open the scoring when Angel Correa scuffed his spot kick wide nine minutes into the second half.
Antony Lozano converted the second of two penalties for Honduras in the 75th minute and they held on to their advantage until the third minute of stoppage time when Mauricio Martinez levelled from a free kick.
Germany edge Sweden to win gold
Argentina, who did not qualify for London 2012 after winning the competition in 2004 and 2008, finished behind Honduras and group winners Portugal in the standings. The Portuguese drew 1-1 with Algeria.
Six more games will be played later on Wednesday to decide the other quarter-finalists.
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