The dollar was given a central place in the international exchange system under the Bretton Woods agreement, which came just after the end of World War II, as you guessed. Before the Great Depression, I don't think there was an analogue, because reserves mostly consisted of gold. However, part of the reason that gold became the monetary standard (as opposed to say silver) was because of its use by the British Empire, which was the largest player in international trade.
On a horribly tangential side-note, if given the chance, have a stay at the Mount Washington Hotel in Bretton Woods, New Hampshire, where they actually hammered all this stuff out. It is of another era, and you can feel the weight of that event in the air there still.
