Let $A$ be an $A_\infty$-algebra over a field $k$. It is a well-known fact that $H^\bullet (A)$ also has an $A_\infty$-structure, and further one can construct a quasi-isomorphism $H^\bullet (A) \to A$. See the notes by Bernhard Keller, Introduction to $A_\infty$ Algebras and Modules.
A similar result is true if you consider instead modules over $A$, i.e., for each $A_\infty$-module $M$ over $A$, the graded space$H^\bullet M$ admits an $A_\infty$-module structure over $A$ and there exists a quasi-isomorphism $H^\bullet M \to M$.
I am looking for a "hands on" proof of the claimed fact about modules. Indeed the justification in Keller's notes is that every quasi-isomorphism is a homotopy equivalence, and so the derived category $\mathcal{D}_\infty A$ is equal to the homotopy category. I (for personal reasons) would like to know if a proof of the quasi-isomorphism could be given directly.
Thanks in advance.