Timeless rival Federer was even faster, dismissing Ukrainian qualifier Alexandr Dolgopolov 6-4, 6-2 in 61 minutes.
Federer, the Swiss world number two, plays Richard Gasquet in the last eight on Friday after Gasquet outlasted fellow Frenchman Gilles Simon in a marathon three-setter.
"I don't know how my wrist is after I fell down", del Potro said before leaving for a local hospital.
"But if nothing risky happened, I will be here to play tomorrow and doing my best".
In the quarterfinals, Nadal will play sixth-seeded Grigor Dimitrov of Bulgaria, who got past Sam Querrey of the US 6-3, 7-6 (3).
Fourth seed Cilic has reached the last four without dropping a set and overcame Nadal's countryman Albert Ramos-Vinolas 6-3 6-4, although the Spaniard did managed to break Cilic's serve for the first time in the competition at 3-3 in the second set.
Tamim ruled out of second Test
On the positives, it seems South Africa's dominant performance leaves them with little to worry about going into the second Test. Morkel has been ruled out for up to six weeks with a side strain sustained in the first-Test victory at Potchefstroom.
That isn't the only time his fun has been spoiled by the Spaniard, either, with Dimitrov reaching his first Grand-Slam semi-final at the start of the year at the Australian Open - only for Nadal to be waiting, again defeating him by a single set.
This year's French Open and US Open champion did not have it all his own way at the Association of Tennis Professionals Masters 1000 tournament, winning 6-3, 6-7, 6-3.
"It was a very tough match".
Trailing 3-2 in the decider, Dimitrov's forehand crashed into the net to hand Nadal the break he needed.
"Dimitrov has got inside Nadal's head".
In the third set, Nadal saved the one break point he faced in the match in the fifth game, and broke serve in the sixth to set up victory.
Dimitrov, chasing a spot at the season-ending ATP Finals in London, beat American Sam Querrey to set up a Chinese reunion with Nadal.