The most important aspect of an answer to this question is that the brassiere was invented well before World War I and existed contemporaneously with the corset.

From about 1895 through 1927, there was a continuous loosening and streamlining of the overall lines of fashion and construction techniques - if you compare day dresses of comparable original values made in just 1895 and 1905, you're generally going to see a less sturdy lining in the second and overall lighter fabrics in general; the exterior fabric of the bodice is going to be cut in a different shape than the lining so that it doesn't look as structured and appears to float over the body, and the skirt and sleeves probably aren't lined at all. Very likely there are fewer different fabrics used and ornamentation is with lace ruffles or overlays rather than heavy beads. Likewise, from 1905 to 1915 you can see the skirt lose its train, become shorter, and go from a complex shape with multiple gores or tucks to something like an A-line, while sleeves stopped being made to curve with the arm and were instead basically symmetrical, front to back.

Corsets likewise were changing over this period. From 1895 to 1905, the whole thing started to move down the body - the fashionable bustline became lower (accentuated by the bloused cut of the bodice) and the corset became more focused on flattening the stomach and giving a curve to the hips and backside. By 1915, again, corsets rarely even supported the bust at all! (Even after this time, the fact that corsets were eventually deemed "inessential" in America and the UK and their materials diverted to the war effort had very little effect on whether or not women actually wore them - it just meant that they couldn't purchase new ones for a few years. Pop history has made a lot out of it but there's little to suggest that it had a large knock-on effect.) The brassiere was actually invented to deal with this in the first decade of the twentieth century, and was at the time more like a slightly shaped cotton crop-top worn over the corset to give the bust a bit of support without raising or defining it. Far from being a replacement for the corset at the time, it was an add-on - and often they were made with a tab at the bottom of the front to attach it to the corset tightly to create more tension for more support.

It is not accurate to say that the early twentieth century killed the corset: the girdle industry boomed during this time. Even during the 1920s, when women were in theory celebrating their natural figures, it failed to die. (I also have to disagree that the lithe 1920s silhouette came up from the lower classes and couldn't make inroads with the upper classes - again, if you go back to the late 1900s/early 1910s, this is very clearly a style emanating from high fashion; actresses were declaring at that time that they didn't need to wear corsets and avant-garde designers were producing informal clothing like Mariano Fortuny's "Delphos dress", a slim sheath of tightly pleated silk that would cling to the wearer's uncorseted body and show every natural flaw.) It's difficult to imagine given the popular narrative of youth and freedom, but much of that narrative has been driven by stereotype. The new twentieth-century girdle was much touted as more comfortable and flexible than the corset due to the use of elastic and spiral steel, but it was still effectively a corset. In the 1930s and 1940s, they were still commonly worn, whether as lower-torso shapewear or as bra-and-girdle-in-one corselets. One major advancement in the 1950s was the popularization of a strapless bra attached to a short corset/girdle (now usually called a "Merry Widow"), which led to and/or was driven by the popularization of the iconic strapless or halter-top evening dresses of the period.

What really killed the corset was the 1960s. Where the 1920s tends to be oversimplified into a quick narrative about the flappers as a youth movement, the 1960s saw a serious counterculture develop that some pioneering designers were quick to get on with. An extreme minimalism of cut, material, and ornamentation took place, with the most extreme version being the one-use-only paper dress. Wearing anything but the lightest kind of bra as a support garment was irredeemably square. Older and more conservative women would continue to make use of structured shapewear, but it would never again achieve the kind of ordinariness that it had had previously.