I am sorry if this question is too trivial but I couldn't find the answer.

Who did first classify topological principal $G$-bundles for some topological group $G$? So, I mean that equivalence classes of principal $G$-bundles on a nice space $X$ are 1-to-1 to homotopy classes [X,BG] where $BG$ denotes the classifying space of $G$.

My first guess was that this is due to Steenrod but then I looked up that Milnor did his construction on classifying spaces in 1956 what is much later.