I would say that, roughly speaking, the Cauchy-Riemann equations are the necessary and sufficient condition for a continously differentiable function to correspond via the definition to an analytic function.
So, basically, the Cauchy-Riemann equations tell us that we can identify a subset of the continously differentiable functions , namely the subset of those that additionally satisfy the Cauchy-Riemann equations, with the analytic functions
A bit devoid of details: Function f analytic in an open set S only if C-R equations hold at every point of S.
Now can we "strengthen" this relation, i.e., what can be said about the converse relation?
I think the closest we can get to the converse is by adding an additional hypothesis.
(Perhaps it's obvious; but if not, here it is.)
The additional hypothesis is continuity of the first partial derivatives of u and v.
So then, the closest we can get to a converse of the implication given in my first post is the following:
Function f is analytic in an open set S if the first partial derivatives of f are continuous and satisfy the C-R equations at every point of S.
(Clearly, the "addition" is mediated through conjunction.)
As far as I can see, that's as close as you can get to the sought after relationship mentioned in your OP.
It's not quite a logical equivalence relation, but it's fairly close.