Schwartzman, the 24th seed from Argentina, won the opening set 6-4. But Australian Ebden, who is ranked 31 in the world, served 12 aces to Schwartzman's one, battled back to win the second 6-3.
But Davis Cup star Schwartzman went through the gears in the next two sets, coming up with big shots at the right time and converting seven out of 14 break points to progress 6-3, 6-2 in a match lasting two hours and nine minutes.
Schwartzman will play Dominik Koepfer of Germany in round two.
Noah Rubin was eliminated on Tuesday, after a four-set defeat against Brit Jay Clarke.