I don't think this attribution makes sense without some further qualification. There is diagram chasing ten years earlier in Cartan-Eilenberg's Homological Alge

I don't think this attribution makes sense without some further qualification. There is diagram chasing ten years earlier in Cartan-Eilenberg's Homological Algebra (1955) -- they prove the five lemma early on -- and Mac Lane mentions the term in his Homology (1963). I wouldn't be surprised if it appears in Eilenberg-Steenrod, or, as you say, Steenrod's earlier work or the Cartan Seminar notes (which I don't have near me right now).