Serbia had a reasonably effective army for their size, the Austrians made questionable strategic decisions, and the Russians provided an invaluable distraction.

The Serbian army of WW1 was fairly large for a nation of their size, had gained valuable experience in the recent Balkan Wars, and was ably led by my Reddit namesake, Field Marshal Radomir Putnik. They also had a geographical and strategic advantage in that they didn't need to invade and defeat the Habsburgs, only to defend their territory from behind a significant river barrier and with favorable terrain for defense. When fighting only against the Austrians, the Serbs had a fairly advantageous position.

Conversely, the Austrians had a less skilled commander, Oskar Potiorek, and suffered from indecisive strategy. Pre-war planning originally assigned a small force to merely pin down the Serbs while the bulk of their army fought the more numerous and threatening Russians in Galicia. However, at the last moment the Austrian commander Hotzendorf decided to switch his Second Army from Galicia to Serbia and seek a quick decision there as well (for mostly political considerations.) WW1 mobilization schedules were very complex and difficult even in the hands of proficient states like Germany, and left to the rather lackadaisical Habsburg bureaucracy, the system was not able to respond to such a last-second change. The result was that the Second Army spent the first phase of the war riding around in trains instead of fighting, and by the time they got anywhere, both the initial invasion of Serbia and the Galician battles had already been decided.

The Austrians did attempt an initial invasion of Serbia with the forces they had available. The Serbs allowed them to advance across the border until they were strung out and far from their supply lines, then struck them in the flank. The Serbs fought well, and the Austrians fought just as well as they did against the Russians, which is to say not at all. The Austrians were thrown back to their own territory, while at the same time the Russians had inflicted a terrible defeat on the Austrians in Galicia and driven all the way to the Carpathian mountains. From this time until the entry of the Bulgarians to the war and the defeat of the Russians in Poland, the Austrians could never afford to concentrate against the Serbs as the Russian and later Italian fronts took up all their available troops. A few latter attempts by the Austrians to invade with limited forces were also fairly easily repulsed.

By the time the Bulgarians arrived in fall of 1915, the Russian front had been stabilized with German help, and many more Central powers' troops could be assigned against the Serbs. This was also more of a priority at the time, as eliminating Serbia would establish a land link between Austria and Germany and their relatively new allies in Bulgaria and Turkey. With a more robust force facing them in the north, the Serbs simply didn't have the manpower to oppose both the Austrians and a fairly proficient and sizable Bulgarian army on their flank. (The border with Bulgaria was even longer than the front against the Austrians.) The Serbs were outnumbered and outflanked.

The TL/DR is that the Serbs never had to fight a significant portion of the Austrian army until the Bulgarians arrived, and once they did the Serbs were doomed.

EDIT: Also, watch it with that "petty nation" business!