Ranked 19 in the world, Argentina Davis Cup star Schwartzman cruised into a two-set lead (6-1, 6-3) before his opponent, Rudolf Molleker of Germany, hit back to win the third set 4-6.
But Schwartzman, who stands at 5ft 7”, raised his game to see out the match 6-0 in the fourth.
Schwartzman will play American Denis Kudla in the next round.
Camila Giorgi of Italy was a first-round winner in the Ladies Singles. The 27th seed beat Dalila Jakupovic of Slovenia 6-3, 6-0.
Giorgi will take on Iga Swiatek of Poland, ranked 178, in round two.