Like DuxBelisarius said, what would become known as the Sudetenland was actually a part of several other realms, mostly Bohemia and Moravia, two regions that have existed in roughly their current borders for centuries basically to this day, and were distinct units – for administrative purposes and the like – even in the various forms of the Habsburg monarchy.
Look at a map showing Europe around the eleventh century, or this map of the Czech crown lands from 1747 or this one from the end of the 19th century to see how largely unchanged these lands remained. So strictly speaking, there was no historical or geographic basis for the territory to "stay" with Austria as it was not simply a part of Austria or a discrete unit to begin with.
What it was, however, was a set of various areas with a large population of Germans. And, very much in the general spirit of 1918 and onwards, that is according to the principles of self-determination and various independent states emerging from the wreckage of Austria-Hungary, these Germans did indeed attempt to break away from the above historical borders.
Following the collapse of the monarchy in November, the four regions of Deutschböhmen, Sudetenland (a smaller territory in the northern part of Moravia, not the later overarching term), Böhmerwaldgau and Deutschsüdmähren declared autonomy and tried aligning themselves with their Austrian cousins in the short lived Deutschösterreich, "German Austria". Which was an entity that claimed the German speaking parts of the former empire not with a view of establishing an independent Austria, but unification with Germany.
This attempt was opposed by the fledgling Czechoslovakia as it tried to preserve the historical borders of its constituent lands for reasons of security and economy. Czechoslovak troops secured the regions in the early days of independence largely unopposed as the Germans were unable or unwilling to force the issue. Their claims continued for several months afterwards, however, for example with plans for German Austrian elections in the above regions, which rather obviously did not take place, or continuing civil unrest and violent protests which sometimes resulted in casualties.
Furthermore and perhaps even more importantly, the victorious powers of the Great War had no desire to see a strengthened Austria, let alone an Austrian province within Germany. Again, like DuxBelisarius said, the Treaty of Saint-Germain-en-Laye saw the above cemented by international support of the Allies. Already in these events of 1919 you can see the beginnings of some of the tensions between Czechs and Germans which would culminate in the Sudetenland during the build up to the Second World War.
During and prior to WWI, the territory referred to as Sudetenland was divided between the Austrian provinces of Bohemia, Moravia, Austrian Silesia, and part of Upper Austria. When the war was reaching it's end in late 1918, Emperor Karl gave autonomy to the Czechs in Bohemia and Moravia; at this stage, the Empire itself was disintegrating, so Czech autonomy was swiftly followed by the formation of a Czech, and later the Czechoslovakian State, which included the Sudetenland, was encompassed the mountainous border regions of the aforementioned Austrian provinces. Sudeten Germans did make some petitions to the Allied Powers for autonomy or incorporation with the Austrian or German state, this would have undermined Czech security by stripping them of a defensible border, and would also have left them without some of the industries there with which their economy could function. Allied loyalty to the Czechs, many of whom had supported the Allies against the Central Powers, and now against the Russian and Hungarian Communists, as well as realpolitik considerations (they did not want to strengthen Austria or Germany to the point where they could threaten the Eastern European states, hence also why Austrian unification with Germany was forbidden) in this case trumped self-determination.
