An excerpt from MacLane's "Category theory for the working mathematician" (29-30)
The fundamental idea of representing a function by an arrow first appeared in topology about 1940, probably in papers or lectures by W. Hurewicz on relative homotopy groups; see [1941].
His initiative immediately attracted the attention of R.H. Fox (see Fox [1943]) and N.E. Steenrod, whose [1941] paper used arrows and (implicitly) functors; see also Hurewicz-Steenrod [1941]. The arrow $f: X\to Y$ rapidly displaced the occasional notation $f(X)\subset Y$ for a function. It expressed well a central interest of topology. Thus a notation (the arrow) led to a concept (category).
Commutative diagrams were probably also first used by Hurewicz.
Categories, functors and natural transformations themselves were discovered by Eilenberg-Mac Lane [1942a] in their study of limits (via natural transformations) for universal coefficient theorems for Čech cohomology. In this paper commutative diagrams appeared in print (probably for the first time). Thus Ext was one of the first functors considered. A direct treatment of categories in their own right appeared in Eilenberg- Mac Lane [1945]...
This last mentioned paper is the one referred to in KConrad's answer.