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, and Murray broke down in tears when he finished off a gritty and determined Del Potro.
The crowd on Centre Court were boisterous throughout, with plenty of Argentine support for Del Potro, although emotions did spill over near the climax when two fans were ejected