Edmund - the current British number two - had been hoping to cause a surprise against the former US Open winner but while he held his own for large periods of the match, he eventually lost out 7-5 6-4.
After the opening six games were shared, Del Potro claimed the first break and although Edmund struck back when he served for the first set, he broke again in the next game before moving closer to the last 16.
The Argentine established a 5-2 lead during the second set but Edmund retrieved one of the breaks when Del Potro attempted to see out the match.
However, despite squandering two match points, Del Potro converted on the third to seal his success in one hour and 45 minutes.
Del Potro will face Kei Nishikori for a spot in the quarter-finals in Rome.