He hit 20 winners as Nadal's winning streak of 37 sets at Roland Garros, going back to his 2015 quarter-final defeat by Novak Djokovic, was ended.
Top seed Halep will take on either German 12th seed Angelique Kerber or Caroline Garcia, the seventh-seeded Frenchwoman who made the last-eight in 2017.
The enforced interlude had clearly benefited Nadal, and assuming he gets the two points he needs to level the match he will start a strong favourite on Thursday. Schwartzman saw two set points come and go but at the third time of asking, after a delay while a spectator was helped out after feeling ill, the Argentine ripped a forehand victor.
But a seismic shock looked on the cards when Schwartzman, who has not beaten Nadal in five previous attempts, won the first set.
Nadal made 14 unforced errors and struck just four winners in a poor first set, before having both of his wrists strapped by the trainer.More news: Natural disaster hits Kilauea summit, sending ash plumes up to 8,000 feet
They will return on Thursday morning.
However, when the players came back on after a delay of nearly an hour Nadal immediately broke back to level the second set.
When play was abandoned under storm clouds about the time most Parisians head for their evening meal, Nadal had repaired much of the damage inflicted on him in the first set, which Schwartzman won 6-4 in and hour-and-a-quarter.
Next up for the Spaniard is a last-eight meeting with Diego Schwartzman, who came from two sets down to see off Kevin Anderson, runner-up to Nadal in last year's US Open final.
He also took his run of consecutive completed sets at the French Open to 34, just seven shy of the record held by Bjorn Borg between 1979 and 1981.