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The Serb, who lost to Del Potro in the bronze-medal play-off at London 2012, was again overpowered by the Argentine, who won 7-6 (7-4) 7-6 (7-2).
The world number one, 29, was in tears as he left the court after the match, which lasted two and a half hours.
"This is one of the toughest losses in my career," said Djokovic, who was trying to win his first Olympic title.
Djokovic became the first man for 47 years to hold all four Grand Slam titles by winning the French Open in June.
He suffered a shock third-round loss to American Sam Querrey at Wimbledon but won his 30th Masters title in Toronto last week.
"It's not easy to handle, especially now, just after the wounds are still fresh," he added after failing to improve on the bronze he won at the 2008 Olympics.
"But you have to deal with it. It's not the first or the last time that I have lost a tennis match. But the Olympic Games, yeah, it's completely different."
Del Potro will face Portugal's Joao Sousa in the second round on Monday.
The 27-year-old, a former US Open winner, has dropped to 141st in the world after two years of injury problems but did not face a single break point on his serve against the 12-time Grand Slam champion.
"It was a wonderful evening from the beginning," said Del Potro.
"After all the effort I've put in to get back to playing tennis, I've defeated the number one. It was a dream night."
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Top seed Novak Djokovic was beaten in the first round of the Olympic men's singles by Juan Martin del Potro.