Mayer beat Albert Ramos of Spain 6-1, 6-3 in just over an hour.
The Argentine later said conditions were tough because of the summer heat but that "I felt great inside the court."
De Bakker defeated Dmitry Tursunov of Russia by 6-3, 6-3.
The Dutchman broke twice for 4-1 lead in the first set and broke early in the second. He also saved all six break points.
Results from the first day of the Croatia Open:
Leonardo Mayer (ARG) bt Albert Ramos (ESP) 6-1, 6-3
Thiemo De Bakker (NED) bt Dmitry Tursunov (RUS) 6-3, 6-3