Del Potro let two break-back points slip in the sixth game and, although his cause looked doomed, he forced Nadal to save three times for 5-3.
The deciding fifth set was a tight battle, but in the fifth game, Nadal got the solitary break to lead 3-2.
Next up for the Spaniard is a blockbuster last four showdown with Novak Djokovic.
Del Potro, who'd saved four set points, grabbed one of his own on Nadal's serve and levelled at a set apiece with a forehand that clipped the net and fell inside the service box.
Del Potro repelled them both but Rafa held at 15 in the following game, clinching the set 6-4 and sending this wonderful encounter into a decider.
Nadal worked the angles and moved superbly from the start, but the heavy-hitting Del Potro frustrated him by saving the first two break-point opportunities of the match before holding to level at 4-4.
Nadal has 17 while Djokovic has stalled on 12 since completing the career Grand Slam at Roland Garros in 2016.
Then, at 4-3, they played an epic 18-point game that featured six deuces and another trio of break points for del Potro, none of which he managed to convert.
"It was a very emotional match. great quality tennis, especially in the last set there were some unbelievable points". Del Potro broke and served for the set at 5-4, but double-faulted and Nadal clinched the break-back with a rasping forehand down the line.
"In the last set there was a little of everything, great points, great rallies, he was hitting insane with his forehands". In some ways he deserves to win as well. I tried to resist and maintain focus.
While you were watching the World Cup semi-final last night, and before that, while you were desperately trying to get out of work to go home and watch the World Cup semi-final, there was a day for ages playing out at Wimbledon, featuring, of course, Roger Federer and Rafa Nadal.
And Nadal rued failing to take control against del Potro much earlier.
Del Potro too noted the change in conditions but appeared to relish the chance to slide around a bit more on the grass.
"I think I won the best point of the match", the Argentine said with a rueful smile after being beaten by Nadal for the 11th time in 16 meetings.