Delbonis won the only previous meeting between the pair in Stuttgart earlier this year, but it was Monaco that triumphed at the ATP 250 event in China, pulling off a 6-3 7-5 success.
Ranked 39 places above his countryman at 58 in the world, many will have expected Delbonis to come through the match with little difficulty.
Instead it was Monaco who was the more solid player throughout as he won 83 per cent of points on his first serve.
The world number 97 also saved all three of the break points against him and will now face Canada's Vasek Pospisil or Russian Teymuraz Gabashvili in the next round.
Italy's Simone Bolelli joins Monaco in the second round of the hard-court tournament following his comfortable victory against Matthew Ebden of Australia.
Bolelli needed just an hour and two minutes to seal his progress and displayed ruthlessness on break points, winning each of the three he earned to qualify 6-4 6-2.
In Monday's other first-round encounter, Chinese wildcard Gao Xin was beaten 6-3 4-6 6-4 by India's Somdev Devvarman.