Schwartzman, 26, had come through a five-set marathon against Marton Fucsovis of Hungary. The Davis Cup star Schwartzman, who is ranked 20th at Roland Garros, held his nerve in an epic match lasting three hours and 41 minutes.
In round two, Schwartzmanbroke serve to win the opening set 6-4. But the big-serving Mayer raised his game to win the next three sets - 6-3, 6-4, 7-5, in a performance that included 15 aces to Schwartzman's four.
Julia Glushko, 29, failed to qualify for the Ladies Singles competition. Having won three qualifying matches, the Israeli was defeated at the round of 24 stage.
Camila Giorgi of Italy withdrew three days before the start of the event.