But Del Potro showed character to break straight back and, after needing treatment on his knee injury, powered his way to three straight games and the set.
When Murray dropped his serve in the opening game of the fourth set it looked like the ascendancy was with Del Potro, who looked to have shaken off his knee problem.
The match again swung, though, with Murray breaking back and then surviving a nervous service game to go 3-2 up.
Both players showed signs of nerves, mixing unforced errors with top-class winners, and they exchanged breaks once more.
Murray kept up the pressure and forced Del Potro to stay in the match, which he failed to do and sent a backhand wide to confirm his fate and seal Murray's passage into the semi-finals.
There he will take on Nadal in a repeat of their Wimbledon quarter-final.
On that occasion Nadal swept Murray aside in straight sets and, although the 22-year-old has since usurped Roger Federer as world number one, Murray's recent form would point to a close encounter.
Nadal went into this quarter-final having never lost a set in four meetings with Fish but that record would soon change as the American raced into a 3-0 lead.
Backed by a partisan home crowd on Arthur Ashe Stadium, Fish carried out his promise of coming into the net whenever possible and taking the game to Nadal.
But Nadal did not take kindly to losing the first set and, with two breaks of serve, raced through the second in 36 minutes.
Fish showed flurries of brilliance in the third but he could not cope with the ferocity of Nadal's groundstrokes and, when he came in to attack, he tended to be passed as the Spaniard found his finest touch.
With the match passing 0200 local time, Nadal was keen to bring it to a close and in the seventh game of the fourth set he broke Fish to go 5-2 ahead.
He then closed out the match comfortably to secure a place in the last four.
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