Spaniard Nadal, bidding for an 11th title on the Parisian clay, also trailed by a break in the second set as Schwartzman threatened a huge shock, but after a rain stoppage, Nadal returned to court to move 5-3 ahead in the second set.
That game had a 40-15 lead for Schwartzman, a brilliant Nadal drop volley that rubbed out the second set point, a Nadal break point, a handsome duo of Schwartzman overheads for a third set point and, just then, after a first-serve fault, a spectator in the 15th row apparently fainting.
Their last-eight match was level at 6-6 (5/5) in a first-set tie-break on Court Suzanne Lenglen after Del Potro had earlier missed five break points.
At one point Nadal, the reigning champion and 10 times victor, found himself a set and a break down to the diminutive Argentinian Schwartzman.
The 25-year-old took his third set point with a rasping forehand up the line, after being made to wait when a spectator fell ill in the stands.
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Nadal then fought back to lead the second set 5-3, before heavy rain brought a halt to the last-eight encounter on Court Philippe Chatrier.
They will return on Thursday morning.
Muguruza condemned Sharapova to her heaviest Grand Slam defeat in more than six years and will face top seed Simona Halep for a place in the final against either Sloane Stephens or Madison Keys.
Soon, with Nadal serving at 3-3, Schwartzman, who did make the 2017 US Open quarterfinals and the 2018 Australian Open fourth round, played the kind of point that kept wreaking the gasps: one on which Nadal appeared the lesser aggressor which, of course, looked weird.
Halep won in 59 minutes with the second set taking just 22 minutes.