Seeded 20, Schwartzman, 27, has started the grand slam tournament with great intent, breezing away his opponents in the first two rounds.
He started the competition with a 6-3, 7-6, 6-0 win over Robin Haase of the Netherlands. This was followed by another three-set victory, this time over Egor Gerasimov from Belarus.
His reward is a match against Tennys Sandgren of America, ranked 72 in the world. Schwartzman's run at Flushing Meadows has taken his tournament earnings to $163,000.
In the Men's Doubles, American duo Noah Rubin and Thai-Son Kwiatkowski lost in three sets against Luke Bambridge and Ben McLachlan.
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