If we view minimizing cross entropy as equivalent to maximizing the log-likelihood of the same model, then I believe we can go as far back as RA Fisher. This places the date between 1912 and 1922, depending on how well-developed you wish the theory to be; see discussion in
John Aldrich "R. A. Fisher and the Making of Maximum
Likelihood 1912 – 1922" Statistical Science. 1997, Vol. 12, No. 3, 162-176
We also have this closely related thread: Definition and origin of “cross entropy” which appears to use the term "cross entropy" in the broad sense of a family of probabilistic losses, instead of the sense used in this post, as jargon for a specific loss for a model of binary data.