Fernando's answer tells you how to prove the statement directly. Alternatively, if you want a reference, this is proven in Corollary 3.6 of my PhD thesis paper (published in JPAA) Model Structures on Commutative Monoids in General Model Categories. I introduce an axiom that a monoidal model category $M$ can satisfy, the "strong commutative monoid axiom," which guarantees that:
- Commutative monoids in $M$ inherit a model structure transferred from $M$ along the forgetful functor $U$, meaning that a morphism $f$ is a weak equivalence or fibration if and only if $U(f)$ is in $M$, and
- $U$ preserves cofibrations with cofibrant source.
Note that (1) implies immediately that $U$ preserves fibrant objects. Then, in Section 5.1, I verify that the example you mention does satisfy this axiom. Furthermore, the initial CDGA is cofibrant, so a corollary of (2) is that $U$ takes cofibrant CDGAs to cofibrant chain complexes (sometimes said "$U$ preserves cofibrant objects").