With world number one Rafael Nadal waiting for the victor, Tuesday's fourth set, played in much cooler conditions than the first three the day before, turned into a mini-epic before Del Potro eventually prevailed 7-6 (1) 7-6 (5) 5-7 7-6 (5).
There was just the one break point in the opening set and it proved to be decisive, the unseeded Vesely gifting Nadal a 3-1 lead with a costly double fault.
The French Open champion will next face Novak Djokovic in a clash of two former champions following the Serbian's success over Kei Nishikori earlier on Wednesday.
Then Del Porto managed to hold serve to love, forcing Nadal to serve out the match, which he did, winning it 6-4 to reach his sixth semi-final.
There was, however, one consolation for Del Potro who came off second best despite belting down 33 aces and producing 77 staggering winners, 10 more than Nadal.
'In the third set he started to play better from the baseline and I had couple of mistakes with the game I lost.
Jiri Vesely went the same way as Nadal's three previous opponents in being dispatched in straight sets and given the Spaniard exited at this stage past year in an epic with Gilles Muller that should not be underestimated. Del Potro, however, opening his shoulders with his feared forehand and blasting a victor that left Nadal gasping on the baseline - and a set down just short of three hours.
Nadal soon returned the favour with a break of his own, before another hold left Del Potro rattled.
Del Potro put both hands on his head.
On this occasion, the world No 1 had trailed two sets to one after a super serving performance in the third and fourth sets by Del Potro.
There was a look of glee in Del Potro's sometimes sad eyes, and his lumbering between-points walk had a bit more energy to it. Nadal, struggling to get a solid second serve into hard areas, was de-fanged for long stretches.
Nadal broke for 3-2 but had to save break points in the sixth and eighth games before securing a memorable victory. Eventually, things went to a tiebreak Del Potro won to capture the set.