As far as I remember this is explained (maybe in differet terms) in Kapranov's paper
"On DG-modules over the de rham complex and the vanishing cycles functor", Lecture Notes in Mathematics, 1991, Volume 1479.
You could also have a look at the discussion below this entry of the everything seminar.
I think the kind of statement you are expecting ("equivalences between carefully defined versions of their "derived categories") are of a type you can find in this paper.
In any way, the equivalence is given by a $\otimes$-Hom adjunction. I might have misunderstood something but I don't really understand where is the problem with having $Hom(\Omega,-)$ and $\Omega\otimes -$.
EDIT: more precisely, I have the feeling that the statement is the following.
If $A$ is Koszul and if $B$ is its Koszul dual then $Hom(A,-)$ and $A\otimes-$ defines an equivalence of DG categories between
$B$-modules
the category $\mathcal P_A$ as it is defined e.g. in Block's paper.
