Delbonis was in no mood to hang around in a 6-4 6-1 triumph in Stuttgart, and the Argentine needed just 80 minutes for the win against his compatriot at the ATP 250 clay-court tournament.
Monaco - runner-up in 2012 - had eight break-point opportunities during the match, but could only convert one of them due to Delbonis' solid defence.
In contrast, eighth seed Delbonis took four of the nine chances to break that came his way, and he also won 76 per cent of first-serve points in a clinical display.
Delbonis' countryman Mayer faced a seemingly easier match on paper against German wildcard Michael Berrer, but he had to fight for a 2-6 6-3 7-6 (7-5) success.
The world number 47 made a poor start and was broken twice in the first set, but quickly recovered to level the match and force a decider.
Neither man was able to find the break in the third, necessitating a tie-break, though Mayer saw four opportunties to do so come and go at 3-3.
Berrer had a mini-break at 3-1 in the decider, but Mayer recovered and clinched the win with his first match point to set up a second-round tie with second seed Mikhail Youzhny.
Andrey Golubev will also contest round two after the Kazakh defeated wildcard Philipp Petzschner 6-4 6-1.