The British number one won the European Open in Antwerp on Sunday for his first trophy on the main tour and followed it up with a 6-3 7-6 (3) win against the Argentinian in Vienna.
It was a tough draw for Edmund, who at 14 is ranked only three places above Schwartzman, and there was very little to choose between the two.
The first set was decided by Edmund's key break of serve at 4-3 and that ended up being the only break point of the match.
Both men dominated on serve in the second set until it went to a tiebreak, where Edmund's winning mentality shone through, taking it 7/3 to seal a straight-sets win.
It was his 11th victory from his last 13 matches and sets up a second-round meeting with Spain's Fernando Verdasco.
Elsewhere, Mikhail Kukushkin clinched his first top-10 victory in four years, battling past fellow Vienna debutant Grigor Dimitrov 6-4 4-6 6-4.
Borna Coric beat Lucas Pouille 4-6 6-0 6-4 and Marton Fucsovics eliminated seventh seed Fabio Fognini 4-6 6-3 6-2 to reach the third round.
There were also second-round victories for Gael Monfils and Karen Khachanov.
Swiss Indoors Open: Zverer's milestone
In Basel, Alexander Zverev became the first player to earn 50 victories this season with a 6-4 7-5 success over Robin Haase.
The German broke twice in the opener, including in the 10th game to take the set after leading 4-1.
Haase was also 4-1 down in the second set before Zverev closed the match out to set up a meeting with 19-year-old Australian qualifier Alexei Popyrin.
Stefanos Tsitsipas, fresh from becoming the first Greek to win an ATP World Tour title in Stockholm, won his 40th match of 2018 to oust France's Jeremy Chardy 6-2 7-6 (3).
Latvia's Ernests Gulbis beat Jack Sock for the second time in six days, following on from last week's Stockholm quarter-final victory, 7-5 6-4.
Spaniard Roberto Bautista Agut knocked out Joao Sousa 6-4 6-3.
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