Let $\pi$ be a group and $K(\pi,1)$ be the associated Eilenberg-MacLane space (it has a contractible universal cover). I have proved the following isomorphism for the the chain complex with local coefficients in some abelian group $A$ with trivial right $\pi$-action:
$$C_*(K(\pi,1);A)\cong A\otimes_{\mathbf Z[\pi]}C_*(\widetilde{K(\pi,1)})$$
I wish to conclude that
$$H_*(K(\pi,1);A)\cong\operatorname{Tor}_*^{\mathbf Z[\pi]}(A,\mathbf Z).$$
I know that the group homology of $\pi$ comes from $\mathbf Z\otimes_{\mathbf Z[\pi]}A$ in some way and I feel like the fact that the universal cover of the space being contractible should allow me to conclude something along similar lines, but I have no good reference for this material and am still fuzzy on how $\operatorname{Tor}$ would work here. I think I basically just want to compute the homology of the chain complex $A\otimes_{\mathbf Z[\pi]}C_*(\widetilde{K(\pi,1)})$ (without using Künneth), but I am confused about how this would work and I'd really appreciate any help.