EDIT: So my attempt at manipulating forms was a failure, so let's just go by general non-sense.
These are CDGA's in the $j^{-1}\mathcal{O}_{X}$-module category (after pulling back $A^\bullet_X(\log D))$ ), so we really just need to check that the quasi-isomorphism preserves the multiplication structure. This can be checked locally, but in local coordinates it's more-or-less obvious that the wedge of logarithmic forms maps to the wedge of the forms restricted to $X \setminus D$.