The Argentinian saved two break points in game eight but he fell short just before the tie break, allowing Nadal to convert a break point in game 12 for a 7-5. Both players have settled into a nice rhythm in the opening set and it was Nadal who kept a high level in the entire set, serving at 75% and losing only eight points on serve, avoiding deuces or tie breaks to keep the pressure on Delpo.
Nadal will face Novak Djokovic next in a rematch of the 2011 final, the last time the Spaniard had even made the quarterfinals at the All England Club.
Former US Open champion Del Potro then stunned the Spaniard with the only break of the third set in the 10th game, secured with a deep forehand.
There's no doubt Djokovic will be the fresher of the two when they meet on Friday, after Del Potro took a lot out of Nadal.
The dogfight Nadal feared and Del Potro knew was his only route to the semi-finals had begun in earnest.
Argentina's Juan Martin Del Potro celebrates winning his the fourth round match against France's Gilles Simon. Nadal was 2-1 down after three sets, but rallied to force a decider, before closing out a 7-5, 6-7 (7), 4-6, 6-4, 6-4 win.
Del Potro was straining to stay in touch in almost every point, and his gambling instincts drove him to explore the white lines.
"Against Rafa", del Potro said, "you must go for winners all the time".
Nadal maintained his typically aggressive approach and was leaping while punching the air after a blistering return whistled past Vesely to put him a break up at 3-2 in the second set. He went 4-3 in front, before closing out the set to increase the pressure on Nadal.
Isner has yet to be broken at this year's tournament and saved the only break point he faced on Wednesday.
As he served out to reach his sixth semi-final, his old sparring partner, Andy Murray, newly installed in the BBC commentary box, declared the quality of tennis in the fifth set "the best I've seen in the tournament so far".