The South Africans seem to have problems holding onto a lead as they let it slip in the highest scoring match of the tournament.
SA finished last in their pool and will now be involved in the classification playoffs for the wooden spoon against India.
They started positively as they opened the scoring seven minutes into the match courtesy of a Justin-Reid Ross goal from a penalty corner.
Their second goal was scored less than a minute later by Lloyd Norris-Jones from a short corner.
At this point the South Africans looked comfortably in the driving seat before the Argentine resurgence started.
The South Americans powered themselves into the match with two goals in the last 10 minutes of the first half to draw the scores level going into the half-time break.
Gonzalo Peillat opened their scoring with a goal from a short corner in the 25th minute while Lucas Rossi scored the equaliser with a field goal.
The Argentines took the match away from the South Africans as they scored three more goals in the first 15 minutes of the second half to give them a comfortable lead.
South Africa struggled to get into the circle, receiving only three penalty corners, converting only one of the scoring opportunities from the short corner.
Argentina on the other hand had five penalty corners of which they converted three of their opportunities.
South Africa launched a gutsy fightback with seven minutes left in the match with Jonty Robinson scoring to narrow Argentina's lead to two goals.
Their efforts, however, were all in vain as the opposition scored another goal by Peillat for his second of the match in the 65th minute.
The SA hockey men have not won an Olympic match since the 2008 Beijing Games and will hope to finally secure a positive result against India.
Olympic men's field hockey results on Tuesday:
Group A
Australia 7 Pakistan 0
Argentina 6 South Africa 3
Group B
South Korea 2 Netherlands 4
India 0 Belgium 3