The last-16 tie between Del Potro and Frenchman Simon was suspended overnight due to darkness on Monday with the Argentine fifth seed two sets to one ahead.
The third set produced more high-quality tennis and Nadal was the first who crumbled under pressure, losing serve at love at 4-5 to drop another set and face an elimination.
Del Potro threw everything he could at Nadal in a pulsating match and even conjured the shot of the tournament as he acrobatically dived to his right to hit a fizzing forehand volley that ended a spellbinding 23-shot rally.
Del Potro, whose best run at the All England Club saw him reach the semi-finals in 2013, triumphed 7-6 (7/1) 7-6 (7/5) 5-7 7-6 (7/5) to set up a last-eight meeting with Rafael Nadal. Simon failed to convert a break point, then a third match point for Del Potro ended with him netting a forehand.
The two-time victor will meet Novak Djokovic in the last four on Friday.
"I think it was great quality tennis and in the final set there were some awesome points", said 2008 and 2010 champion Nadal.
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If they had beaten Croatia , England would have reached only their second World Cup final. "We were anxious ", Katich said. Croatia opened at 60-1 World Cup odds to win the entire tournament; entering the semis, it was still the underdog at 4-1.
Soon enough, del Potro was down on his belly after one last slip, as Nadal finished off the victory with a volley. He saved another break point to inch closer to the semi-final, but just couldn't put Del Potroaway during a terrific and gruelling fifth set.
"Rafa is a fighter", del Potro said.
In the next set, Del Potro broke and served for the set at 5-4, but double-faulted and Nadal clinched the break-back with a forehand victor down the line for the second set to be decided by a tiebreaker.
The deciding fifth set was a tight battle, but in the fifth game, Nadal got the solitary break to lead 3-2.
"In the end the court was a little bit slippery because the sun goes out and it was humid on the grass", Del Potro added.
Anderson, who reached his first Grand Slam final at the 2017 U.S. Open, had lost in the fourth round at Wimbledon on three previous occasions.