This is the earliest reference I have located:

<A HREF="https://cowles.yale.edu/sites/default/files/files/pub/cdp/m-0416.pdf">Minkowski-Farkas Lemma in Banach Spaces</A>, L. Hurwicz (1952).

The same result was also referred to as the Minkowski-Farkas-Weyl theorem in the 1950's, for example in 

<A HREF="https://pubmed.ncbi.nlm.nih.gov/16590290/">The strong Minkowski-Farkas-Weyl theorem for vector spaces over ordered fields</A>, A. Charnes and W.W. Cooper (1958).

The three authors are associated with this result in view of the following contributions:

* H. Minkowski, *Geometrie der Zahlen* (Leipzig, 1896).
* J. Farkas, "*Theorie der einfachen Ungleichungen*," J. F. Reine u. Ang. Mat., **124**, 1-27 (1902).
* H. Weyl, "*Elementare Theorie der konvexen Polyeder*," Comm. Helvet., **7**, 290-306 (1935).