Is there an explicit way to invert a quasi-isomorphism of two chain complexes?
In case of homotopy algebras ($A_\infty$, $L_\infty$ ect.) there is an explicit way to invert any quasi isomorphism, if we are willing to work with infinity-morphisms. The inversion is then basically done by the homotopy transfer theorem.
Is something like this available for plain chain complexes?
Just think of $\Bbb Z/2 \Bbb Z$ and of $\Bbb Z \xrightarrow{\cdot 2} \Bbb Z$ as chain complexes, so that both only have non-vanishing homology in degree zero.
The the quotient map $\Bbb Z \to \Bbb Z/2 \Bbb Z$ induces a quasi isomorphism, but obviously there is no nontrivial map in the other direction.
Though what you could do is find some kind of projective resolution of the chain complexes and then the induced map you will get should have a quasi inverse. You could look for model categories(the projective model structure on chain complexes) cofibrant replacement and the Whitehead theorem.