Konig's Theorem for Min Weight Vertex Cover?
So I found the answer.

The equivalence is that the min weight vertex cover of a bipartite graph can be computed as the maximum flow in a related bipartite graph. In the unweighted case, this maximum flow corresponds to the maximum carnality matching in a bipartite which is exactly the version of Konig's theorem that we all know and love.

For the sake of completeness here is the reduction of bipartite min weight vertex cover to max flow.

Let $G = (A,B)$ be the given bipartite graph. Construct a flow network $N$  by connecting the a source $S$ to all nodes in $A$ and all nodes in $B$ to a sink $T$.

Let the capacity of all original edges in $G$ be $\infty$. The capacity of all edges of the form $(S,a)$ where $a \in A$ will be $w_a$ and the similarly the capacity of all edges $(b, T)$ where $b \in B$ will be $w_b$. Every $S-T$ cut in this network corresponds to exactly one vertex cover and every vertex cover corresponds to an $S-T$ cut. Thus the min-cut a.k.a. the maximum flow will give us the minimum weight vertex cover.

For a more detailed proof refer this.