An emotional Murray, 29, secured a thrilling 7-5 4-6 6-2 7-5 victory to make it GB's most successful day at an overseas Games with five gold medals.
The see-saw match featured 14 breaks of serve in front of a raucous crowd.
Murray's win comes five weeks after he claimed his second Wimbledon title and four years after London 2012 success.
Both men looked exhausted in the fourth set of a gruelling four-hour contest, with Murray breaking down in tears after finishing off his gritty opponent.
Find out how to get into tennis in our special guide.
The crowd on Centre Court were boisterous throughout, with plenty of Argentine support for 27-year-old Del Potro, although emotions did spill over near the climax when two fans were ejected.
Murray said the final was one of the toughest he had played in and cast doubt on the chances of winning a third gold in Tokyo in 2020.
"Four years is a long time and so many things can change," he said. "Who knows about Tokyo? At 33, I'm not sure I'll be at the same level."
Del Potro's silver medal signifies his continued return to form, with his past three seasons disrupted by a wrist injury.
"Mentally I can only imagine how frustrating that must have been to keep going through the same problem and having to try and come back," said Murray.
"He deserves a lot of credit and he should be very proud."
Del Potro beat world number one Novak Djokovic in the first round despite being stuck in a lift for 40 minutes before the match.
The 2009 US Open champion and London 2012 bronze medallist then defeated 2008 champion Rafael Nadal in an epic three-hour semi-final a day before facing Murray.
"I left the last of my toenails on the court," said Del Potro. "I was tremendously tired. It was the crowd that made me keep running."
He added: "When I look at my medal I'm so happy for that, but I've just lost a great final against Andy and I was close to the gold medal.
"He played better in the important moments and is a great champion."
In Murray, Del Potro came up against a player on a career-defining run in the final.
After inspiring Britain to a first Davis Cup win in 79 years in November, Murray was voted the BBC's Sports Personality of the Year, before reaching the Australian and French Open finals and winning Wimbledon for the second time.
He also became a father earlier this year and led GB out at the opening ceremony of Rio 2016 as his country's flag bearer.
On the court, he has lost just one of his past 30 matches, a run stretching back six months, and has been world number two since November, barring a one-week drop to third in May.
BBC tennis correspondent Russell Fuller
Murray looked all along as if he would be too good and just too strong, but he had to play a quite brilliant game to prevent the match going to a deciding set.
Some exceptional defence and some awesome returns allowed him to break back for five-all - and two games later he was a double Olympic champion.
He may look back on this achievement as the greatest of his career.
It will mean even more to him that he shared it with Max Whitlock, Justin Rose and Jason Kenny on such a special day for Team GB.
Earlier, Japan's Kei Nishikori claimed his first Olympic medal by beating Spain's 14-time Grand Slam winner Nadal for men's singles bronze.
Nadal, 30, who had already won men's doubles gold, recovered from 5-2 down in the second set to level the match, but lost 6-2 6-7 (1-7) 6-3.
Nishikori, 26, had never been past the quarter-finals at a Games.
Britain's Andy Murray became the first male tennis player to win two Olympic singles titles by beating Argentina's Juan Martin del Potro in Rio.