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.
The American qualifer won the first set 6-4 on Court 8, but the home crowd got behind Clarke, who served 15 aces, and won 7-5, 6-4, 6-4.
Rubin's frustrations got the better of him upon losing the third set and he received a code violation. Clarke will face Roger Federer in round two.
Camila Giorgi was eliminated by Dayana Yastremska on Monday.