In a match lasting just over two hours on Court Rainier III in Monaco, Schwartzman fought back in the second set as he won nine consecutive games to turn the match around.
Edmund, who was knocked out of last week's Grand Prix Hassan II in Marrakech by Jo-Wilfried Tsonga at the second round stage, took the first set in 50 minutes against his Argentinian opponent, clinching it 6-4 in the 10th game against serve.
The 17th-seeded Brit raced to a 3-0 lead in the second set but Schwartzman hit back to level matters at 3-3 before going on to claim it 6-3.
The Argentine continued to dominate as he won the first three games of the third set - to extend his winning streak to nine games - before Edmund briefly stopped the rot to make it 3-1, but Schwartzman won the next three games to seal the win.
Next up for Schwartzman is a clash with the winner of the match between Tsonga and Taylor Fritz which is first on court on Tuesday.
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Daniil Medvedev advanced to the second round following a comfortable straight-sets victory over Joao Sousa.
The 10th seed thrashed his Portuguese opponent 6-1 6-1 in just 55 minutes to set up a clash with Radu Albot, who beat qualifier Aljaz Bedene 6-4 6-2.
Marco Cecchinato, seeded 11th, progressed after Damir Dzumhur retired when 4-0 down in the first set, while fellow Italian Fabio Fognini was made to work hard for a comeback win over qualifier Andrey Rublev.
The 13th seed fought back from losing the first set to seal a 4-6 7-5 6-4 triumph.
David Goffin, who is seeded 16th, saw off Guido Andreozzi 6-1 6-4 but 12th seed Nikoloz Basilashvili was beaten 7-5 3-6 6-1 by Marton Fucsovics.
Denis Shapovalov, seeded 15, was another high-profile casualty as he was knocked out by Jan-Lennard Struff 5-7 6-3 6-1.
There were also wins for Roberto Bautista Agut, who set up a tie with 11-time winner and defending champion Rafael Nadal, qualifier Lorenzo Sonego and Spaniard Jaume Munar.
Elsewhere, Dusan Lajovic, Philipp Kohlschreiber, who faces top seed Novak Djokovic next, Mikhail Kukushkin and Martin Klizan also advanced.
